Octonions and Triality

Complex numbers and quaternions form special cases of lower-dimensional Clifford algebras, their even subalgebras and their ideals

\begin{matrix}ℂ ≃ Cℓ_{0,1} ≃ Cℓ_{2}^{+} ≃ Cℓ_{0,2}^{+}, \\ ℍ ≃ Cℓ_{0,2} ≃ Cℓ_{3}^{+} ≃ Cℓ_{0,3}^{+} ≃ \frac{1}{2}(1 ± e_{123})Cℓ_{0,3}. \end{matrix}

In this chapter, we explore another generalization of ℂ and ℍ, a non-associative real algebra, the Cayley algebra of octonions, 𝕆. Like complex numbers and quaternions, octonions form a real division algebra, of the highest possible dimension, 8. As an extreme case, 𝕆 makes its presence felt in classifications, for instance, in conjunction with exceptional cases of simple Lie algebras.

Like ℂ and ℍ, 𝕆 has a geometric interpretation. The automorphism group of ℍ is SO(3), the rotation group of ℝ^{3} in ℍ = ℝ ⊕ ℝ^{3}. The automorphism group of 𝕆 = ℝ ⊕ ℝ^{7} is not all of SO(7), but only a subgroup, the exceptional Lie group G_{2}. The subgroup G_{2} fixes a 3-vector, in ⋀^{3} ℝ^{7}, whose choice determines the product rule of 𝕆.

The Cayley algebra 𝕆 is a tool to handle an esoteric phenomenon in dimension 8, namely triality, an automorphism of the universal covering group Spin(8) of the rotation group SO(8) of the Euclidean space ℝ^{8}. In general, all automorphisms of SO(n) are either inner or similarities by orthogonal matrices in O(n), and all automorphisms of Spin(n) are restrictions of linear transformations Cℓ_{n} → Cℓ_{n}, and project down to automorphisms of SO(n). The only exception is the triality automorphism of Spin(8), which cannot be linear while it permutes cyclically the three non-identity elements −1, e_{12...8}, −e_{12...8} in the center of Spin(8).

We shall see that triality is a restriction of a polynomial mapping Cℓ_{8} → Cℓ_{8}, of degree 2. We will learn how to compose trialities, when they correspond to different octonion products. We shall explore triality in terms of classical linear algebra by observing how eigenplanes of rotations transform under triality.

23.1 Division algebras An algebra A over ℝ is a linear (that is a vector) space A over ℝ together with a bilinear map A×A → A, (a, b) → ab, the algebra product. Bilinearity means distributivity (a + b)c = ac + bc, a(b + c) = ab + ac and (λa)b = a(λb) = λ(ab) for all a, b, c ∈ A and λ ∈ ℝ. An algebra is without zero-divisors if ab = 0 implies a = 0 or b = 0. In a division algebra 𝔻 the equations ax = b and ya = b have unique solutions x, y for all non-zero a ∈ 𝔻. A division algebra is without zero-divisors, and conversely, every finite-dimensional algebra without zero-divisors is a division algebra. If a division algebra is associative, then it has unity 1 and each non-zero element has a unique inverse (on both sides).

An algebra with a unity is said to admit inverses if each non-zero element admits an inverse (not necessarily unique). An algebra is alternative if a(ab) = a^{2}b and (ab)b = ab^{2}, and flexible if a(ba) = (ab)a. An alternative algebra is flexible. An alternative division algebra has unity and admits inverses, which are unique. The only alternative division algebras over ℝ are ℝ, ℂ, ℍ and 𝕆.

An algebra A with a positive-definite quadratic form N: A → ℝ, is said to preserve norm, or admit composition, if for all a, b ∈ A, N(ab) = N(a)N(b). The dimension of a norm-preserving division algebra 𝔻 over ℝ is 1, 2, 4 or 8; if furthermore 𝔻 has unity, then it is ℝ, ℂ, ℍ or 𝕆.

Examples. 1. Define in ℂ a new product a∘b by a∘b = ab̄. Then ℂ becomes a non-commutative and non-alternative division algebra over ℝ, without unity.

2. Consider a 3-dimensional algebra over ℝ with basis {1, i, j} such that 1 is the unity and i^{2} = j^{2} = −1 but ij = ji = 0. The algebra is commutative and flexible, but non-alternative. It admits inverses, but inverses of the elements of the form xi + yj are not unique, (xi + yj)^{−1} = λ(yi − xj) − \frac{ix + jy}{x^{2} + y^{2}} , where λ ∈ ℝ. It has zero-divisors, by definition, and cannot be a division algebra, although all non-zero elements are invertible.

3. Consider a 3-dimensional algebra over ℚ with basis {1, i, i^{2}}, unity 1 and multiplication table

\begin{aligned}& i & i^{2} \\ i & i^{2} & 3 \\ i^{2} & 3 & −6i \end{aligned}

The algebra is commutative and flexible, but non-alternative. Each non-zero element has a unique inverse. Multiplication by x + iy + i^{2}z has determinant x^{3} + 3y^{3} − 18z^{3}, which has no non-zero rational roots (Euler 1862). Thus, the algebra is a division algebra, 3D over ℚ.

23.2 The Cayley–Dickson doubling process Complex numbers can be considered as pairs of real numbers with componentwise addition and with the product

(x_{1}, y_{1})(x_{2}, y_{2}) = (x_{1}x_{2} − y_{1}y_{2}, x_{1}y_{2} + y_{1}x_{2}).

Quaternions can be defined as pairs of complex numbers, but this time the product involves complex conjugation

(z_{1}, w_{1})(z_{2}, w_{2}) = (z_{1}z_{2} − w_{1}w̄_{2}, z_{1}w_{2} + w_{1}z̄_{2}).

Octonions can be defined as pairs of quaternions, but this time order of multiplication matters

(p_{1}, q_{1}) ∘ (p_{2}, q_{2}) = (p_{1}p_{2} − q̄_{2}q_{1}, q_{2}p_{1} + q_{1}p̄_{2}).

This doubling process, of Cayley-Dickson, can be repeated, but the next algebras are not division algebras, although they still are simple and flexible (Schafer 1954). Every element in such a Cayley-Dickson algebra satisfies a quadratic equation with real coefficients.

Example. The quaternion q = w+ix+jy+kz satisfies the quadratic equation

q^{2} − 2wq + |q|^{2} = 0.

The Cayley-Dickson doubling process

\begin{matrix}ℂ = ℝ ⊕ ℝi \\ ℍ = ℂ ⊕ ℂj \\ 𝕆 = ℍ ⊕ ℍℓ \end{matrix}

provides a new imaginary unit ℓ, ℓ^{2} = −1, which anticommutes with i, j, k. The basis {1, i, j, k} of ℍ is complemented to a basis {1, i, j, k, ℓ, iℓ, jℓ, kℓ} of 𝕆 = ℍ ⊕ ℍℓ. Thus, 𝕆 is spanned by 1 ∈ ℝ and the 7 imaginary units i, j, k, ℓ, iℓ, jℓ, kℓ, each with square −1, so that 𝕆 = ℝ ⊕ ℝ^{7}. Among subsets of 3 imaginary units, there are 7 triplets, which associate and span the imaginary part of a quaternionic subalgebra. The remaining 28 triplets anti-associate.

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The multiplication table of the unit octonions can be summarized by the Fano plane, the smallest projective plane, consisting of 7 points and 7 lines, with orientations. The 7 oriented lines correspond to the 7 quaternionic/associative triplets.

23.3 Multiplication table of 𝕆 303

23.3 Multiplication table of 𝕆 Denote the product of a, b ∈ 𝕆 by a ∘ b. Let 1, e_{1}, e_{2}, …, e_{7} be a basis of 𝕆. Define the product in terms of the basis by

e_{i} ∘ e_{i} = −1, and e_{i} ∘ e_{j} = −e_{j} ∘ e_{i} for i ≠ j,

and by the table

e_{1} ∘ e_{2} = e_{4}, e_{2} ∘ e_{4} = e_{1}, e_{4} ∘ e_{1} = e_{2},
e_{2} ∘ e_{3} = e_{5}, e_{3} ∘ e_{5} = e_{2}, e_{5} ∘ e_{2} = e_{3},
⋮  ⋮  ⋮
e_{7} ∘ e_{1} = e_{3}, e_{1} ∘ e_{3} = e_{7}, e_{3} ∘ e_{7} = e_{1}.

The table can be condensed into the form

e_{i} ∘ e_{i+1} = e_{i+3}

where the indices are permuted cyclically and translated modulo 7.

If e_{i} ∘ e_{j} = ±e_{k}, then e_{i}, e_{j}, e_{k} generate a subalgebra isomorphic to ℍ. The sign in e_{i} ∘ e_{j} = ±e_{k} can be memorized by rotating the triangle in the following picture by an integral multiple of 2π/7 :

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; e_{7}; e_{1}; e_{2}; e_{3}; e_{4}; e_{5}; e_{6}; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .]

Example. The product e_{2} ∘ e_{5} = −e_{3} corresponds to a triangle obtained by rotating the picture by 2π/7.

In the Clifford algebra Cℓ_{0,7} of ℝ^{0,7}, octonions can be identified with paravectors, 𝕆 = ℝ ⊕ ℝ^{0,7}, and the octonion product may be expressed in terms of the Clifford product as

a ∘ b = ⟨ab(1 − v)⟩_{0,1},

where v = e_{124} + e_{235} + e_{346} + e_{457} + e_{561} + e_{672} + e_{713} ∈ ⋀^{3} ℝ^{0,7}. In Cℓ_{0,7}, the octonion product can be also written as

a ∘ b = ⟨ab(1 + w)(1 − e_{12...7})⟩_{0,1} for a, b ∈ ℝ ⊕ ℝ^{0,7}

where \frac{1}{8}(1 + w)\frac{1}{2}(1 − e_{12...7}) is an idempotent, w = ve_{12...7}^{−1} ∈ ⋀^{4} ℝ^{0,7} and e_{12...7}^{−1} = e_{12...7}.

In the Clifford algebra Cℓ_{8} of ℝ^{8}, we represent octonions by vectors, 𝕆 = ℝ^{8}. As the identity of octonions we choose the unit vector e_{8} in ℝ^{8}. The octonion product is then expressed in terms of the Clifford product as

a ∘ b = ⟨ae_{8}b(1 + w)(1 − e_{12...8})⟩_{1} for a, b ∈ ℝ^{8}

where \frac{1}{8}(1 + w)\frac{1}{2}(1 − e_{12...8}) is an idempotent, w = ve_{12...7}^{−1} ∈ ⋀^{4} ℝ^{8}, e_{12...7}^{−1} = −e_{12...7} and v = e_{124} + e_{235} + e_{346} + e_{457} + e_{561} + e_{672} + e_{713} ∈ ⋀^{3} ℝ^{8}.

23.4 The octonion product and the cross product in ℝ^{7} A product of two vectors is linear in both factors. A vector-valued product of two vectors is called a cross product, if the vector is orthogonal to the two factors and has length equal to the area of the parallelogram formed by the two vectors. A cross product of two vectors exists only in 3D and 7D.

The cross product of two vectors in ℝ^{7} can be constructed in terms of an orthonormal basis e_{1}, e_{2}, …, e_{7} by antisymmetry, e_{i} × e_{j} = −e_{j} × e_{i}, and

e_{1} × e_{2} = e_{4}, e_{2} × e_{4} = e_{1}, e_{4} × e_{1} = e_{2},
e_{2} × e_{3} = e_{5}, e_{3} × e_{5} = e_{2}, e_{5} × e_{2} = e_{3},
⋮  ⋮  ⋮
e_{7} × e_{1} = e_{3}, e_{1} × e_{3} = e_{7}, e_{3} × e_{7} = e_{1}.

The above table can be condensed into the form

e_{i} × e_{i+1} = e_{i+3}

where the indices are permuted cyclically and translated modulo 7. This cross product of vectors in ℝ^{7} satisfies the usual properties, that is,

(a × b) · a = 0, (a × b) · b = 0  orthogonality
|a × b|^{2} = |a|^{2}|b|^{2} − (a · b)^{2}  Pythagorean theorem

where the second rule can also be written as |a × b| = |a||b| sin ∢(a, b). Unlike the 3-dimensional cross product, the 7-dimensional cross product does not satisfy the Jacobi identity, (a × b) × c + (b × c) × a + (c × a) × b ≠ 0, and so it does not form a Lie algebra. However, the 7-dimensional cross product satisfies the Malcev identity, a generalization of Jacobi, see Ebbinghaus et al. 1991 p. 279.

In ℝ^{3}, the direction of a × b is unique, up to two alternatives for the orientation, but in ℝ^{7} the direction of a × b depends on a 3-vector defining the 23.4 The octonion product and the cross product in ℝ^{7} 305

cross product; to wit,

a × b = −(a ∧ b) v [≠ −(a ∧ b)v]

depends on v = e_{124} + e_{235} + e_{346} + e_{457} + e_{561} + e_{672} + e_{713} ∈ ⋀^{3} ℝ^{7}. In the 3-dimensional space a × b = c × d implies that a, b, c, d are in the same plane, but for the cross product a × b in ℝ^{7} there are also other planes than the linear span of a and b giving the same direction as a × b.

The 3-dimensional cross product is invariant under all rotations of SO(3), while the 7-dimensional cross product is not invariant under all of SO(7), but only under the exceptional Lie group G_{2}, a subgroup of SO(7). When we let a and b run through all of ℝ^{7}, the image set of the simple bivectors a ∧ b is a manifold of dimension 2·7−3 = 11 > 7 in ⋀^{2} ℝ^{7}, dim(⋀^{2} ℝ^{7}) = \frac{1}{2}7(7−1) = 21, while the image set of a × b is just ℝ^{7}. So the mapping

a ∧ b → a × b = −(a ∧ b) v

is not a one-to-one correspondence, but only a method of associating a vector to a bivector.

The 3-dimensional cross product is the vector part of the quaternion product of two pure quaternions, that is,

a × b = Im(ab) for a, b ∈ ℝ^{3} ⊂ ℍ.

In terms of the Clifford algebra Cℓ_{3} ≃ Mat(2, ℂ) of the Euclidean space ℝ^{3} the cross product could also be expressed as

a × b = −⟨abe_{123}⟩_{1} for a, b ∈ ℝ^{3} ⊂ Cℓ_{3}.

In terms of the Clifford algebra Cℓ_{0,3} ≃ ℍ×ℍ of the negative definite quadratic space ℝ^{0,3} the cross product can be expressed not only as

a × b = −⟨abe_{123}⟩_{1} for a, b ∈ ℝ^{0,3} ⊂ Cℓ_{0,3}

but also as ^{1}

a × b = ⟨ab(1 − e_{123})⟩_{1} for a, b ∈ ℝ^{0,3} ⊂ Cℓ_{0,3}.

Similarly, the 7-dimensional cross product is the vector part of the octonion product of two pure octonions, that is, a × b = ⟨a ∘ b⟩_{1}. The octonion algebra 𝕆 is a norm-preserving algebra with unity 1, whence the vector part ℝ^{7} in 𝕆 = ℝ ⊕ ℝ^{7} is an algebra with cross product, that is, a × b = \frac{1}{2}(a ∘ b − b ∘ a) for a, b ∈ ℝ^{7} ⊂ 𝕆 = ℝ ⊕ ℝ^{7}. The octonion product in turn is given by

a ∘ b = αβ + αb + aβ − a · b + a × b

1 This expression is also valid for a, b ∈ ℝ^{3} ⊂ Cℓ_{3}, but the element 1 − e_{123} does not pick

up an ideal of Cℓ_{3}. Recall that Cℓ_{3} is simple, that is, it has no proper two-sided ideals.

for a = α + a and b = β + b in ℝ ⊕ ℝ^{7}. If we replace the Euclidean space ℝ^{7} by the negative definite quadratic space ℝ^{0,7}, then not only

a ∘ b = αβ + αb + aβ + a · b + a × b

for a, b ∈ ℝ ⊕ ℝ^{0,7}, but also

a ∘ b = ⟨ab(1 − v)⟩_{0,1}

where v = e_{124} + e_{235} + e_{346} + e_{457} + e_{561} + e_{672} + e_{713} ∈ ⋀^{3} ℝ^{0,7}.

23.5 Definition of triality Let n ≥ 3. All automorphisms of SO(n) are of the form U → S US^{−1} where S ∈ O(n). All automorphisms of Spin(n), n ≠ 8, are of the form u → sus^{−1} where s ∈ Pin(n). The group Spin(8) has exceptional automorphisms, which permute the non-identity elements −1, e_{12...8}, −e_{12...8} in the center of Spin(8) :

−1 ⟶ e_{12...8}
↖ ↙ ρ(±e_{12...8}) = −I.
−e_{12...8}

Such an automorphism of Spin(8), of order 3, is said to be a triality automorphism, denoted by trial(u) for u ∈ Spin(8).

Regard Spin(8) as a subset of Cℓ_{8}. In Cℓ_{8}, triality sends the lines through 1, −e_{12...8} and −1, e_{12...8}, which are parallel, to the lines through 1, −1 and e_{12...8}, −e_{12...8}, which intersect each other. Thus, a triality automorphism of Spin(8) cannot be a restriction of a linear mapping Cℓ_{8} → Cℓ_{8}.

A non-linear automorphism of Spin(8) might also interchange −1 with either of ±e_{12...8}. Such an automorphism of Spin(8), of order 2, is said to be a swap automorphism, denoted by swap(u) for u ∈ Spin(8).

On the Lie algebra level, triality acts on the space of bivectors ⋀^{2} ℝ^{8}, of dimension 28. Triality stabilizes point-wise the Lie algebra G_{2} of G_{2}, which is the automorphism group of 𝕆. The dimension of G_{2} is 14. In the orthogonal complement G_{2}^{⊥} of G_{2}, triality is an isoclinic rotation, turning each bivector by the angle 120^{∘}. A swap stabilizes point-wise not only G_{2} but also a 7dimensional subspace of G_{2}^{⊥}, and reflects the rest of the Lie algebra so(8) ≃ D_{4}, that is, another 7-dimensional subspace of G_{2}^{⊥}. For a bivector F ∈ ⋀^{2} ℝ^{8}, we denote triality by Trial(F) and swap by Swap(F).

On the level of representation spaces, triality could be viewed as permuting the vector space ℝ^{8} and the two even spinor spaces, that is, the minimal left ideals Cℓ_{8}^{+} \frac{1}{8}(1 + w)\frac{1}{2}(1 ± e_{12...8}), which are sitting in the two-sided ideals Cℓ_{8}^{+} \frac{1}{2}(1±e_{12...8}) ≃ Mat(8, ℝ) of Cℓ_{8}^{+} ≃^{2}Mat(8, ℝ). This means a 120^{∘} rotation of the Coxeter–Dynkin diagram of the Lie algebra D_{4} :

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Rather than permuting the representation spaces, triality permutes elements of Spin(8), or their actions on the vector space and the two spinor spaces.

Because of its relation to octonions, it is convenient to view triality in terms of the Clifford algebra Cℓ_{0,7} ≃^{2}Mat(8, ℝ), the paravector space $R^{8} = ℝ⊕ℝ^{0,7}, having an octonion product, the spin group

$pin(8) = {u ∈ Cℓ_{0,7} | uū = 1; for all x ∈ $R^{8} also uxû^{−1} ∈ $R^{8}},

the minimal left ideals Cℓ_{0,7} \frac{1}{8}(1 + w)\frac{1}{2}(1 ∓ e_{12...7}) of Cℓ_{0,7} ≃^{2}Mat(8, ℝ), and the primitive idempotents

f = \frac{1}{8}(1 + w)\frac{1}{2}(1 − e_{12...7}), f̂ = \frac{1}{8}(1 + w)\frac{1}{2}(1 + e_{12...7}).

For u ∈ $pin(8), define two linear transformations U_{1}, U_{2} of $R^{8} by

U_{1}(x) = 16⟨uxf⟩_{0,1}, U_{2}(x) = 16⟨uxf̂⟩_{0,1}.

The action of u on the left ideal Cℓ_{0,7} \frac{1}{8}(1 + w) of Cℓ_{0,7} results in the matrix representation ^{2} ^{3}

$pin(8) ∋ u ≃ ( \begin{matrix}U_{1} & 0 \\ 0 & U_{2} \end{matrix} ) where U_{1}, U_{2} ∈ $O(8).

For U ∈ SO(8), ^{4} define the companion Ǔ by

Ǔ(x) = { \atop U(x̂)} for all x ∈ $R^{8}.

u ≃ ( \begin{matrix}U_{1} & 0 \\ 0 & Ǔ_{2} \end{matrix} ) ,

2 Choose the bases (e_{1}f, e_{2}f, …, e_{7}f, f) for Cℓ_{0,7}f and (e_{1}f̂, e_{2}f̂, …, e_{7}f̂, f̂) for Cℓ_{0,7}f̂,

where f = \frac{1}{8} (1 + w) \frac{1}{2} (1 − e_{12...7}) and f̂ = \frac{1}{8} (1 + w) \frac{1}{2} (1 + e_{12...7}). Then the matrices

of U_{1} and U_{2} are the same as in the basis (e_{1}, e_{2}, …, e_{7}, 1) of $R^{8}. Denoting f_{i} = e_{i}f,

i = 1, 2, …, 7, and f_{8} = f, (U_{1})_{ij} = 16⟨f̄_{i}uf_{j}⟩_{0}, and denoting g_{i} = e_{i}f̂, i = 1, 2, …, 7,

and g_{8} = f̂, (U_{2})_{ij} = 16⟨ḡ_{i}ug_{j}⟩_{0}.

3 If we had chosen the bases (f_{1}, f_{2}, …, f_{7}, f) for Cℓ_{0,7}f and (f̂_{1}, f̂_{2}, …, f̂_{7}, f̂) for Cℓ_{0,7}f̂,

where f_{i} = e_{i}f and f̂_{i} = −e_{i}f̂ for i = 1, 2, …, 7, then we would have obtained the

following matrix representation

where U_{1}(x) = 16⟨uxf⟩_{0,1} as before but Ǔ_{2}(x) = 16⟨ûxf⟩_{0,1}. This representation is used

by Porteous 1995.

4 Or, for U ∈ Mat(8, ℝ).

The companion ǔ of u ∈ $pin(8) is just its main involution, ǔ = û, ^{5} and corresponds to the matrix

ǔ ≃ ( \begin{matrix}Ǔ_{2} & 0 \\ 0 & Ǔ_{1} \end{matrix} ) .

For a paravector a ∈ $R^{8}, define the linear transformation A of $R^{8} by ^{6}

A(x) = 16⟨axf⟩_{0,1}, that is, A(x) = a ∘ x,

making $R^{8} the Cayley algebra 𝕆. Since Ǎ^{⊤}(x) = 16⟨axf̂⟩_{0,1}, we have the correspondence

a ≃ ( \begin{matrix}A & 0 \\ 0 & Ǎ^{⊤} \end{matrix} ) , abbreviated as a ∼ A.

Computing the matrix product

U(a) = uaû^{−1} ≃ ( \begin{matrix}U_{1} & 0 \\ 0 & U_{2} \end{matrix} ) ( \begin{matrix}A & 0 \\ 0 & Ǎ^{⊤} \end{matrix} ) ( \begin{matrix}Ǔ_{2}^{−1} & 0 \\ 0 & Ǔ_{1}^{−1} \end{matrix} ) ,

we find the correspondence U(a) ∼ U_{1}AǓ_{2}^{−1}. Denote U_{0} = Ǔ, and let Ǔ_{0}(a) operate on x ∈ $R^{8}, to get

Ǔ_{0}(a) ∘ x = (U_{1}AǓ_{2}^{−1})(x) = U_{1}(a ∘ Ǔ_{2}^{−1}(x)).

The ordered triple (U_{0}, U_{1}, U_{2}) in SO(8) is called a triality triplet with respect to the octonion product of 𝕆.

If (U_{0}, U_{1}, U_{2}) is a triality triplet, then also (U_{1}, U_{2}, U_{0}), (U_{2}, U_{0}, U_{1}) and (Ǔ_{2}, Ǔ_{1}, Ǔ_{0}) are a triality triplets. This results in

Ǔ_{0}(x ∘ y) = U_{1}(x) ∘ U_{2}(y) for all x, y ∈ 𝕆 = $R^{8},

referred to as Cartan’s principle of triality. Conversely, for a fixed U_{0} ∈ $O(8), the identity Ǔ_{0}(x ∘ y) = U_{1}(x) ∘ U_{2}(y) has two solutions U_{1}, U_{2} in $O(8), resulting in the triality triplet (U_{0}, U_{1}, U_{2}) and its opposite (U_{0}, −U_{1}, −U_{2}).

A = ( \begin{matrix}a_{0} & −a_{4} & −a_{7} & a_{2} & −a_{6} & a_{5} & a_{3} & a_{1} \\ a_{4} & a_{0} & −a_{5} & −a_{1} & a_{3} & −a_{7} & a_{6} & a_{2} \\ a_{7} & a_{5} & a_{0} & −a_{6} & −a_{2} & a_{4} & −a_{1} & a_{3} \\ −a_{2} & a_{1} & a_{6} & a_{0} & −a_{7} & −a_{3} & a_{5} & a_{4} \\ a_{6} & −a_{3} & a_{2} & a_{7} & a_{0} & −a_{1} & −a_{4} & a_{5} \\ −a_{5} & a_{7} & −a_{4} & a_{3} & a_{1} & a_{0} & −a_{2} & a_{6} \\ −a_{3} & −a_{6} & a_{1} & −a_{5} & a_{4} & a_{2} & a_{0} & a_{7} \\ −a_{1} & −a_{2} & −a_{3} & −a_{4} & −a_{5} & −a_{6} & −a_{7} & a_{0} \end{matrix} ) .

5 Recall that for x ∈ $R^{8} = ℝ ⊕ ℝ^{0,7}, U(x) = uxû^{−1}, and so Ǔ(x) = ûx̂u^{−1}.

6 The matrix of A can be computed as A_{ij} = 16⟨f̄_{i}af_{j}⟩_{0}. The paravector a = a_{0} + a_{1}e_{1} +

⋯ + a_{7}e_{7} has the matrix

Thus, U_{0} corresponds to two triality triplets (U_{0}, U_{1}, U_{2}) and (U_{0}, −U_{1}, −U_{2}), while −U_{0}, corresponds to (−U_{0}, −U_{1}, U_{2}) and (−U_{0}, U_{1}, −U_{2}).

The rotations U_{1}, U_{2} ∈ $O(8) are represented by ±u_{1}, ±u_{2} ∈ $pin(8). We choose the signs so that

û_{0} ≃ ( \begin{matrix}U_{1} & 0 \\ 0 & U_{2} \end{matrix} ) , û_{1} ≃ ( \begin{matrix}U_{2} & 0 \\ 0 & U_{0} \end{matrix} ) , û_{2} ≃ ( \begin{matrix}U_{0} & 0 \\ 0 & U_{1} \end{matrix} ) ,

where u_{0} = û and U_{0} = Ǔ. Using the notion of triality triplets,

u_{0} ≃ (U_{0}, U_{1}, U_{2}), u_{1} ≃ (U_{1}, U_{2}, U_{0}), u_{2} ≃ (U_{2}, U_{0}, U_{1}).

The rotation U_{0} in $O(8) corresponds to u_{0} ≃ (U_{0}, U_{1}, U_{2}) and its opposite −u_{0} ≃ (U_{0}, −U_{1}, −U_{2}) in $pin(8), and the opposite rotation −U_{0} corresponds to e_{12...7}u_{0} ≃ (−U_{0}, −U_{1}, U_{2}) and −e_{12...7}u_{0} ≃ (−U_{0}, U_{1}, −U_{2}). Triality is defined as the mapping

trial : $pin(8) → $pin(8), u_{1} ≃ ( \begin{matrix}Ǔ_{0} & 0 \\ 0 & Ǔ_{2} \end{matrix} ) → u_{2} ≃ ( \begin{matrix}Ǔ_{1} & 0 \\ 0 & Ǔ_{0} \end{matrix} ) .

Triality is an automorphism of $pin(8); it is of order 3 and permutes the nonidentity elements −1, e_{12...7}, −e_{12...7} in the center of $pin(8).

Example. Take a unit paravector a ∈ $R^{8} = ℝ ⊕ ℝ^{0,7}, |a| = 1. The action x → axâ^{−1} is a simple rotation of $R^{8}. ^{7} Thus, a ∈ $pin(8). Denote a_{0} = â, a_{1} = trial(a_{0}) and a_{2} = trial(a_{1}) so that 16⟨â_{1}xf⟩_{0,1} = a_{2}xâ_{2}^{−1}, 16⟨â_{2}xf̂⟩_{0,1} = a_{1}xâ_{1}^{−1}. Then

a ∘ x = a_{1}xâ_{1}^{−1} and x ∘ a = a_{2}xâ_{2}^{−1}

represent isoclinic rotations of $R^{8}. Left and right multiplications by a ∈ S^{7} ⊂ 𝕆 are positive and negative isoclinic rotations of $R^{8} = 𝕆. ^{8} The Moufang identity

a ∘ (x ∘ y) ∘ a = (a ∘ x) ∘ (y ∘ a)

results in a special case of Cartan’s principle of triality ^{9}

â_{0}(x ∘ y)a_{0}^{−1} = (a_{1}xâ_{1}^{−1}) ∘ (a_{2}yâ_{2}^{−1}).

In this special case, a_{0}, a_{1}, a_{2} commute, a_{2} = ã_{1} (= â_{1}^{−1}) and a_{0}a_{1}a_{2} = 1 implying a = a_{1}a_{2} = a_{1}â_{1}^{−1} = a_{2}â_{2}^{−1}.

7 Note that a ∘ x ∘ a = axã, ã = â^{−1} and a ∘ x ∘ a^{−1} = sxs^{−1} where s = a_{1}a_{2}^{−1} ∈ $pin(7).

8 Any four mutually orthogonal invariant planes of an isoclinic rotation of $R^{8} induce the

same orientation on $R^{8}.

9 To prove Cartan’s principle of triality, in the general case, iterate the Moufang identity,

like b ∘ a ∘ (x ∘ y) ∘ a ∘ b = (b ∘ (a ∘ x)) ∘ ((y ∘ a) ∘ b). Observe the nesting b ∘ (a ∘ x) = sxŝ^{−1},

where s = trial(b̂)trial(â) = trial(b̂a).

Triality sends a simple rotation to a positive isoclinic rotation and a positive isoclinic rotation to a negative isoclinic rotation. The isoclinic rotations can be represented by octonion multiplication having neutral axis in the rotation plane of the simple rotation:

positive isoclinic rotation ↗ simple rotation ↓ ↖

negative isoclinic rotation

23.6 Spin(7) Let u_{0} ∈ Spin(7) ⊂ Cℓ_{8}, and u_{1} = trial(u_{0}), u_{2} = trial(u_{1}). Then u_{2} = ǔ_{1}, that is, trial(trial(u_{0})) = e_{8}trial(u_{0})e_{8}^{−1}. ^{10} Thus, u_{1}ǔ_{2}^{−1} = 1 and u_{1}u_{2}^{−1} = u_{1}e_{8}u_{1}^{−1}e_{8}^{−1} ∈ ℝ ⊕ ℝ^{7}e_{8}, being a product of two vectors, represents a simple rotation. ^{11} ^{12} Since Ǔ_{0} = U_{0}, U_{2} = Ǔ_{1},

u_{0} ≃ ( \begin{matrix}U_{1} & 0 \\ 0 & Ǔ_{1} \end{matrix} ) , u_{1} ≃ ( \begin{matrix}U_{0} & 0 \\ 0 & U_{1} \end{matrix} ) , u_{2} ≃ ( \begin{matrix}U_{2} & 0 \\ 0 & U_{0} \end{matrix} ) .

Comparing matrix entries of u_{1}u_{0}^{−1}u_{2}, we find u_{1}u_{0}^{−1}u_{2} ∈ Spin(7) and so U_{1}U_{0}^{−1}U_{2} ∈ SO(7).

Let the rotation angles of U_{0} ∈ SO(7) be α_{0}, β_{0}, γ_{0} so that α_{0} ≥ β_{0} ≥ γ_{0} ≥ 0. Then the rotation angles of U_{1} ∈ SO(8) are

\{ \begin{matrix}α_{1} = \frac{1}{2}(α_{0} + β_{0} + γ_{0}) \\ β_{1} = \frac{1}{2}(α_{0} + β_{0} − γ_{0}) \\ γ_{1} = \frac{1}{2}(α_{0} − β_{0} + γ_{0}) \\ δ_{1} = \frac{1}{2}(α_{0} − β_{0} − γ_{0}). \end{matrix}

Since eigenvalues change in U_{0} → U_{1}, triality cannot be a similarity, U_{1} ≠ SU_{0}S^{−1}. Represent the rotation planes of U_{0} ∈ SO(7) ⊂ SO(8) by unit bivectors A_{0}, B_{0}, C_{0}, and choose the orientation of D_{0} = ue_{8}, u ∈ ℝ^{7}, |u| = 1 so that A_{0} ∧ B_{0} ∧ C_{0} ∧ D_{0} = e_{12...8}. The rotation planes of U_{1} can be expressed as unit bivectors ^{13}

A_{1} = \frac{1}{2}Trial(A_{0} + B_{0} + C_{0} − D_{0})
B_{1} = \frac{1}{2}Trial(A_{0} + B_{0} − C_{0} + D_{0})
C_{1} = \frac{1}{2}Trial(A_{0} − B_{0} + C_{0} + D_{0})
D_{1} = \frac{1}{2}Trial(A_{0} − B_{0} − C_{0} − D_{0}).

10 Note that trial(trial(u_{1})) ≠ e_{8}trial(u_{1})e_{8}^{−1}, trial(trial(u_{2})) ≠ e_{8}trial(u_{2})e_{8}^{−1}.

11 In Cℓ_{0,7}, u_{2} = û_{1}, and so u_{1}û_{2}^{−1} = 1, but u_{1}u_{2}^{−1} ∈ ℝ ⊕ ℝ^{0,7}.

12 Recall that for u ∈ Spin(8), u^{−1} = ũ, and for u ∈ $pin(8), u^{−1} = ū.

13 Trial : ⋀^{2} ℝ^{8} → ⋀^{2} ℝ^{8} sends negative isoclinic bivectors to simple bivectors.

23.7 The exceptional Lie algebra G_{2} 311

The rotation angles and planes of U_{2} are

α_{2} = α_{1}, β_{2} = β_{1}, γ_{2} = γ_{1}, δ_{2} = −δ_{1}
A_{2} = Ǎ_{1}, B_{2} = B̌_{1}, C_{2} = Č_{1}, D_{2} = −Ď_{1}.

The rotation planes of U_{0}, U_{1}, U_{2} induce the same orientation on ℝ^{8}, that is,

A_{0} ∧ B_{0} ∧ C_{0} ∧ D_{0} = A_{1} ∧ B_{1} ∧ C_{1} ∧ D_{1} = A_{2} ∧ B_{2} ∧ C_{2} ∧ D_{2}.

For u_{0} ∈ Spin(7), u_{1}, u_{2} ∈ Spin(8), so that

u_{0} = exp(\frac{1}{2}(α_{0}A_{0} + β_{0}B_{0} + γ_{0}C_{0}))
u_{1} = exp(\frac{1}{2}(α_{1}A_{1} + β_{1}B_{1} + γ_{1}C_{1} + δ_{1}D_{1}))
u_{2} = exp(\frac{1}{2}(α_{1}A_{2} + β_{1}B_{2} + γ_{1}C_{2} − δ_{1}D_{2})).

23.7 The exceptional Lie algebra G_{2} A rotation U ∈ SO(7) such that

U(x ∘ y) = U(x) ∘ U(y) for all x, y ∈ 𝕆

is an automorphism of the Cayley algebra 𝕆. The automorphisms form a group G_{2} with Lie algebra G_{2} ⊂ ⋀^{2} ℝ^{8}, dim G_{2} = 14. A bivector U ∈ G_{2} acts on the octonion product as a derivation

U (x ∘ y) = (U x) ∘ y + x ∘ (U y) for all x, y ∈ 𝕆 = ℝ^{8}.

The double cover of G_{2} ⊂ SO(7) in Spin(7) consists of two components, G_{2} and −G_{2}. The groups G_{2} and G_{2} = ρ(G_{2}) are isomorphic, G_{2} ≃ G_{2}. ^{14}

A rotation U_{0} ∈ G_{2} ⊂ SO(7) has only one preimage in G_{2} ⊂ Spin(7), say u_{0}, ρ(u_{0}) = U_{0}. Since trial(u_{0}) = u_{0}, u_{1} = trial(u_{0}) equals u_{0}, and U_{1} = ρ(u_{1}) equals U_{0}. The rotation angles α_{0}, β_{0}, γ_{0} of U_{0}, such that α_{0} ≥ β_{0} ≥ γ_{0} ≥ 0, satisfy the identities

\{ \begin{matrix}α_{1} = \frac{1}{2}(α_{0} + β_{0} + γ_{0}) = α_{0} \\ β_{1} = \frac{1}{2}(α_{0} + β_{0} − γ_{0}) = β_{0} \\ γ_{1} = \frac{1}{2}(α_{0} − β_{0} + γ_{0}) = γ_{0} \\ δ_{1} = \frac{1}{2}(α_{0} − β_{0} − γ_{0}) = 0 \end{matrix}

each of which implies

α_{0} = β_{0} + γ_{0}.

14 Note that −I ∉ G_{2} because −I ∉ SO(7) and −1 ∉ G_{2} because triality stabilizes point-

wise G_{2} but sends −1 to ±e_{12...8}.

This can also be expressed by saying that the signed rotation angles α, β, γ of U ∈ G_{2} satisfy

α + β + γ = 0.

Represent the rotation planes of U by unit bivectors A, B, C and choose orientations so that u = exp(\frac{1}{2}(αA + βB + γC)), when U = ρ(u). Then A w = B + C. Conversely, for an arbitrary rotation U ∈ SO(7) to be in G_{2} it is sufficient that

A w = B + C and α + β + γ = 0.

In order to construct a bivector U ∈ G_{2}, pick up a unit bivector A ∈ ⋀^{2} ℝ^{7}, A^{2} = −1, decompose the bivector A w into a sum of two simple unit bivectors B+C (this decomposition is not unique), choose α, β, γ ∈ ℝ so that α+β+γ = 0, and write U = αA + βB + γC.

For u ∈ G_{2}, trial(u) = u, and for U ∈ G_{2}, Trial(U) = U, in other words, triality stabilizes point-wise G_{2} and G_{2}. Multiplication by u ∈ G_{2} stabilizes the idempotent \frac{1}{8}(1 + w), u\frac{1}{8}(1 + w) = \frac{1}{8}(1 + w)u = \frac{1}{8}(1 + w), while a bivector U ∈ G_{2} annihilates \frac{1}{8}(1 + w), U\frac{1}{8}(1 + w) = \frac{1}{8}(1 + w)U = 0, and thus U\frac{1}{8}(7 − w) = \frac{1}{8}(7 − w)U = U. Conversely, a rotation U ∈ SO(7) is in G_{2} if it fixes the 3-vector

v = e_{124} + e_{235} + e_{346} + e_{457} + e_{561} + e_{672} + e_{713}

for which w = ve_{12...7}^{−1} = e_{1236} − e_{1257} − e_{1345} + e_{1467} + e_{2347} − e_{2456} − e_{3567}. A bivector F ∈ ⋀^{2} ℝ^{8}, dim(⋀^{2} ℝ^{8}) = 28, can be decomposed as

F = G + H where G ∈ G_{2} and H = \frac{1}{3}w (w ∧ F) ∈ G_{2}^{⊥}.

Under triality, F goes to Trial(F) = G + Trial(H), Trial(H) ∈ G_{2}^{⊥}, where the angle between H and Trial(H) is 120^{∘}. In particular, triality is an isoclinic rotation when restricted to G_{2}^{⊥}, dim(G_{2}^{⊥}) = 14.

A bivector F ∈ ⋀^{2} ℝ^{7} can be decomposed as F = G + H, where G ∈ G_{2},

H ∈ G_{2}^{⊥} ∩ ⋀^{2} ℝ^{7}, dim(G_{2}^{⊥} ∩ ⋀^{2} ℝ^{7}) = 7.

For a vector a ∈ ℝ^{7}, v a ∈ G_{2}^{⊥} ∩⋀^{2} ℝ^{7}. The mapping a → v a is one-to-one, since a = \frac{1}{3}v (v a). The element u = exp(v a) ∈ Spin(7) induces a rotation of ℝ^{7}, which has a as its axis and is isoclinic in a^{⊥} = {x ∈ ℝ^{7} | x · a = 0}. A miracle happens when |a| = 2π/3. Then the rotation angles of U = ρ(u) are 4π/3, which is the same as 4π/3−2π = −2π/3 in the opposite sense of rotation. For the signed rotation angles we can choose α = 4π/3, β = γ = −2π/3 which satisfy α + β + γ = 0. Since also A w = B + C, it follows that u ∈ G_{2}. Therefore, u = exp(v a), where a ∈ ℝ^{7} and |a| = 2π/3, belongs to

exp(G_{2}) ∩ exp(G_{2}^{⊥} ∩ ⋀^{2} ℝ^{7}) ≃ S^{6}, where exp(G_{2}) = G_{2}.

Note that α + β + γ = 0 in G_{2} while α = β = γ in exp(G_{2}^{⊥} ∩ ⋀^{2} ℝ^{7}). An element u ∈ G_{2} ∩ exp(G_{2}^{⊥} ∩ ⋀^{2} ℝ^{7}) can be also constructed by choosing a unit bivector A ∈ ⋀^{2} ℝ^{7}, A^{2} = −1, decomposing A w = B + C, constructing bivectors

\frac{2π}{3} (2A − B − C) = G ∈ G_{2} and \frac{2π}{3} (−A − B − C) = H ∈ G_{2}^{⊥} ∩ ⋀^{2} ℝ^{7}

and exponentiating

u = e^{G} = e^{H} = −\frac{1}{8} + ⋯

The elements u are extreme elements in G_{2} in the sense that ⟨u⟩_{0} = −\frac{1}{8}, while for all other g ∈ G_{2}, ⟨g⟩_{0} > −\frac{1}{8}.

The elements u = exp(v a), where a ∈ ℝ^{0,7} and |a| = 2π/3, satisfy u^{3} = 1, and they are the only non-identity solutions of u^{3} = 1 in G_{2}. The octonion a = e^{∘a} (= e^{a}) satisfies a^{∘3} = 1 and a^{∘−1} ∘ x ∘ a = uxu^{−1} for all x ∈ $R^{8} = ℝ ⊕ ℝ^{0,7}. Conversely, the only unit octonions a ∈ S^{7} ⊂ 𝕆 = $R^{8} satisfying

a^{∘−1} ∘ (x ∘ y) ∘ a = (a^{∘−1} ∘ x ∘ a) ∘ (a^{∘−1} ∘ y ∘ a) for all x, y ∈ $R^{8}

are solutions of a^{∘3} = ±1.

23.8 Components of the automorphism group of Spin(8) In general, the only exterior automorphisms of Spin(n), n ≠ 8, are of type u → sus^{−1}, where s ∈ Pin(n)\Spin(n). Thus, Aut(Spin(n))/Int(Spin(n)) ≃ ℤ_{2}, when n ≠ 8. However, in the case n = 8, the following sequence is exact

1 ⟶ Int(Spin(8)) ⟶ Aut(Spin(8)) ⟶ S_{3},

that is, Aut(Spin(8))/Int(Spin(8)) ≃ S_{3}, a non-commutative group of order 6.

For u ∈ Spin(8), denote swap_{1}(u) = e_{8}trial(u)e_{8}^{−1} = trial(trial(e_{8}ue_{8}^{−1})) and swap_{2}(u) = e_{8}trial(trial(u))e_{8}^{−1} = trial(e_{8}ue_{8}^{−1}). Then trial, swap_{1}, swap_{2} generate S_{3} :

trial ∘ trial ∘ trial = swap_{1} ∘ swap_{1} = swap_{2} ∘ swap_{2} = identity
swap_{1} ∘ swap_{2} = trial and swap_{2} ∘ swap_{1} = trial ∘ trial.

The automorphism group of Spin(8) contains 6 components, represented by the identity, trial, trial ∘ trial, swap_{1}, swap_{2} and the companion. ^{15} In the component of trial, all automorphisms of order 3 are trialities for some octonion product.

23.9 Triality is quadratic Triality of u ∈ Spin(8) ⊂ Cℓ_{8} is a restriction a polynomial mapping Cℓ_{8} → Cℓ_{8}, of degree 2,

trial(u) = trial_{1}(u)trial_{2}(u)
trial_{1}(u) = \frac{1}{2}(1 + e_{12...8})[⟨u(1 + w)(1 − e_{12...8})⟩_{0,6} ∧ e_{8}]e_{8}^{−1}
+\frac{1}{2}(1 − e_{12...8})
trial_{2}(u) = (w − 3)[(u(1 + e_{12...8})) ∧ e_{8}]e_{8}^{−1}(w − 3)^{−1}.

The first factor is affine linear and the second factor is linear. To verify that trial is a triality, it is sufficient to show that it is an automorphism of order 3 sending −1 to e_{12...8}.

In the Lie algebra level, the triality automorphism of a bivector F ∈ ⋀^{2} ℝ^{8} is

Trial(F) = e_{8}⟨F − \frac{1}{2}F(1 + w)(1 + e_{12...8})⟩_{2}e_{8}^{−1}
= \frac{1}{2}e_{8}(F − F w − (F ∧ w) e_{12...8})e_{8}^{−1}.

The triality automorphism of a para-bivector F ∈ ℝ^{0,7} ⊕ ⋀^{2} ℝ^{0,7} is

Trial(F) = ⟨F − \frac{1}{2}F (1 + w)(1 − e_{12...7})⟩_{1,2}^{∧}
= \frac{1}{2}(F − ⟨F ⟩_{2} w + (F ∧ w) e_{12...7})^{∧}.

For u ∈ $pin(8), triality is

trial(u) = trial_{1}(u)trial_{2}(u)
trial_{1}(u) = \frac{1}{2}(1 − e_{12...7})⟨u(1 + w)(1 + e_{12...7})⟩_{0,6}
+\frac{1}{2}(1 + e_{12...7})
trial_{2}(u) = (w − 3) even(u(1 − e_{12...7}))(w − 3)^{−1}.

23.10 Triality in terms of eigenvalues and invariant planes Triality can also be viewed classically, without Clifford algebras, by inspection of changes in eigenvalues and invariant planes of rotations. Consider U_{0} ∈ SO(8) and a triality triplet (U_{0}, U_{1}, U_{2}). Let the rotation angles α_{0}, β_{0}, γ_{0}, δ_{0} of U_{0} be such that α_{0} ≥ β_{0} ≥ γ_{0} ≥ δ_{0} ≥ 0. Represent the rotation planes of U_{0} by the unit bivectors A_{0}, B_{0}, C_{0}, D_{0}. Then the rotation angles of U_{1} and U_{2} are

15 The subgroup of linear automorphisms contains 2 components, represented by the identity

and the companion, u → e_{8}ue_{8}^{−1}.

\{ \begin{matrix}α_{1} = \frac{1}{2}(α_{0} + β_{0} + γ_{0} − δ) \\ β_{1} = \frac{1}{2}(α_{0} + β_{0} − γ_{0} + δ) \\ γ_{1} = \frac{1}{2}(α_{0} − β_{0} + γ_{0} + δ) \\ δ_{1} = \frac{1}{2}(α_{0} − β_{0} − γ_{0} − δ) \end{matrix} and \{ \begin{matrix}α_{2} = \frac{1}{2}(α_{0} + β_{0} + γ_{0} + δ) \\ β_{2} = \frac{1}{2}(α_{0} + β_{0} − γ_{0} − δ) \\ γ_{2} = \frac{1}{2}(α_{0} − β_{0} + γ_{0} − δ) \\ δ_{2} = \frac{1}{2}(−α_{0} + β_{0} + γ_{0} − δ) \end{matrix}

and the rotation planes are

A_{1} = \frac{1}{2}Trial(A_{0} + B_{0} + C_{0} − D_{0})
B_{1} = \frac{1}{2}Trial(A_{0} + B_{0} − C_{0} + D_{0})
C_{1} = \frac{1}{2}Trial(A_{0} − B_{0} + C_{0} + D_{0})
D_{1} = \frac{1}{2}Trial(A_{0} − B_{0} − C_{0} − D_{0})

and

A_{2} = \frac{1}{2}Trial(Trial(A_{0} + B_{0} + C_{0} + D_{0}))
B_{2} = \frac{1}{2}Trial(Trial(A_{0} + B_{0} − C_{0} − D_{0}))
C_{2} = \frac{1}{2}Trial(Trial(A_{0} − B_{0} + C_{0} − D_{0}))
D_{2} = \frac{1}{2}Trial(Trial(−A_{0} + B_{0} + C_{0} − D_{0})).

23.11 Trialities with respect to different octonion products An arbitrary 4-vector w ∈ ⋀^{4} ℝ^{8}, for which \frac{1}{8}(1+w) is an idempotent in Cℓ_{8}^{+}, is called a calibration. ^{16} A direction n ∈ ℝ^{8}, |n| = 1, fixed by the calibration, wn = nw, is called the neutral axis of the calibration. A calibration together with its neutral axis can be used to introduce an octonion product on ℝ^{8} and a triality of Spin(8).

Let w_{1}, w_{2} be calibrations, with neutral axes n_{1}, n_{2} ∈ ℝ^{8}. Denote the octonion products by

x ∘_{w_{1},n_{1}} y = ⟨xn_{1}y(1 + w_{1})(1 − e_{12...8})⟩_{1},
x ∘_{w_{2},n_{2}} y = ⟨xn_{2}y(1 + w_{2})(1 − e_{12...8})⟩_{1}

and the trialities by trial_{w_{1},n_{1}}(u), trial_{w_{2},n_{2}}(u). Denote the opposite of the composition of the trialities by trial_{w_{12},n_{12}}(u) so that

trial_{w_{12},n_{12}}(trial_{w_{12},n_{12}}(u)) = trial_{w_{1},n_{1}}(trial_{w_{2},n_{2}}(u)).

Then

w_{12} = \frac{1}{2}(w_{1} + w_{2}) + \frac{1}{2}(−w_{1} + w_{2})e_{12...8}
= \frac{1}{2}(1 − e_{12...8})w_{1} + \frac{1}{2}(1 + e_{12...8})w_{2}

16 Note that w satisfies w^{2} = 7 + 6w.

and

n_{12} = \frac{y}{|y|} for a non-zero y = x − \frac{1}{4}w_{12} (w_{12} x), where x ∈ ℝ^{8}.

23.12 Factorization of u ∈ Spin(8)

Take u ∈ Spin(8) \ Spin(7). Denote s = (u ∧ e_{8})e_{8}^{−1}, and u_{7} = \frac{s}{|s|}. Then

u_{7} ∈ Spin(7), and u = u_{8}u_{7} = u_{7}u′_{8}, where u_{8}, u′_{8} ∈ ℝ ⊕ ℝ^{7}e_{8}. ^{17} These factorizations are unique, up to a sign (−1 is a square root of 1 in Spin(7)) :

u = u_{8}u_{7} = (−u_{8})(−u_{7}) = u_{7}u′_{8} = (−u_{7})(−u′_{8}).

The following factorizations are unique, up to a cube root of 1 in G_{2} :

u_{7} = h_{0}g_{0} = h_{1}g_{1} = h_{2}g_{2} = g_{0}h′_{0} = g_{1}h′_{1} = g_{2}h′_{2},

where

h_{0}^{3} = u_{7}trial(u_{7})^{−1}u_{7}trial(trial(u_{7}))^{−1},
h_{0}^{′3} = trial(u_{7})^{−1}u_{7}trial(trial(u_{7}))^{−1}u_{7},

and

h_{1} = h_{0}g, h_{2} = h_{0}g^{2}, h′_{1} = g′h′_{0}, h′_{2} = g^{′2}h′_{0},
g = exp(\frac{2π}{3} v \frac{h_{0}}{|h_{0}|}), g′ = exp(\frac{2π}{3} v \frac{h′_{0}}{|h′_{0}|}),
h_{0} = (H_{0} ∧ H_{0} ∧ H_{0})e_{12...7}, h′_{0} = (H′_{0} ∧ H′_{0} ∧ H′_{0})e_{12...7},
h_{0} = e^{H_{0}}, h′_{0} = e^{H′_{0}}.

In this factorization, g_{0}, g_{1}, g_{2} ∈ G_{2} and h_{0}, h_{1}, h_{2}, h′_{0}, h′_{1}, h′_{2} ∈ exp(G_{2}^{⊥}∩⋀^{2} ℝ^{7}) and g, g′ ∈ G_{2} ∩ exp(G_{2}^{⊥} ∩ ⋀^{2} ℝ^{7}) ≃ S^{6}. These factorizations are unique, up to a factor g, g′ ∈ G_{2}, g^{3} = g^{′3} = 1. ^{18}

Appendix: Comparison of formalisms in ℝ^{8} and ℝ ⊕ ℝ^{0,7} We use the 3-vector

v = e_{124} + e_{235} + e_{346} + e_{457} + e_{561} + e_{672} + e_{713}

in ⋀^{3} ℝ^{8} or ⋀^{3} ℝ^{0,7}, and the 4-vector w = ve_{12...7}^{−1} in ⋀^{4} ℝ^{8} or ⋀^{4} ℝ^{0,7},

w = e_{1236} − e_{1257} − e_{1345} + e_{1467} + e_{2347} + e_{2456} − e_{3467}.

Note that e_{12...7}^{−1} = −e_{12...7} in Cℓ_{8} while e_{12...7}^{−1} = e_{12...7} in Cℓ_{0,7}.

17 Note that u_{8} = \sqrt{u(e_{8}ue_{8}^{−1})^{−1}} and u′_{8} = \sqrt{(e_{8}ue_{8}^{−1})^{−1}u}.

18 Recall that −1 ∉ G_{2}.

23.12 Factorization of u ∈ Spin(8) 317

We use the octonion product

x ∘ y = ⟨xe_{8}y(1 + w)(1 − e_{12...8})⟩_{1} for vectors x, y ∈ ℝ^{8}, x ∘ y = ⟨xy(1 + w)(1 − e_{12...7})⟩_{0,1} for paravectors x, y ∈ ℝ ⊕ ℝ^{0,7}.

Note that in Cℓ_{0,7}, also x ∘ y = ⟨xy(1 − v)⟩_{0,1}.

The bivector F = A + B ∈ ⋀^{2} ℝ^{8}, with B ∈ ⋀^{2} ℝ^{7} and A = ae_{8}, a ∈ ℝ^{7}, corresponds to the para-bivector F = a−B ∈ ℝ^{0,7}⊕⋀^{2} ℝ^{0,7}, with a = Ae_{8}^{−1} ∈ ℝ^{0,7}. Let u = u_{+} + u_{−}e_{8} ∈ Spin(8), where u_{±} ∈ Cℓ_{7}^{±}. Then u ∈ Spin(8) corresponds to

({ \atop u_{+} + u_{−}}) = ({ \atop u_{+} + u_{−}})^{−1} ∈ $pin(8).

The companion ǔ of u is

ǔ = e_{8}ue_{8}^{−1}  for u in Spin(8) or Cℓ_{8}^{+} (or Cℓ_{8}),
ǔ = û  for u in $pin(8) or Cℓ_{0,7}.

For u_{0}, u_{1} = trial(u_{0}), u_{2} = trial(u_{1}), Cartan’s principle of triality says

ǔ_{0}(x ∘ y)ǔ_{0}^{−1} = (u_{1}xu_{1}^{−1}) ∘ (u_{2}yu_{2}^{−1})  in Spin(8),
û_{0}(x ∘ y)u_{0}^{−1} = (u_{1}xû_{1}^{−1}) ∘ (u_{2}yû_{2}^{−1})  in $pin(8).

In the Lie algebra level, Freudenthal’s principle of triality says ^{19}

(x ∘ y) F̌_{0} = (x F_{1}) ∘ y + x ∘ (y F_{2}) in ⋀^{2} ℝ^{8},
⟨(x ∘ y) F_{0}⟩_{0,1} = ⟨x F̂_{1}⟩_{0,1} ∘ y + x ∘ ⟨y F̂_{2}⟩_{0,1} in ℝ^{0,7} ⊕ ⋀^{2} ℝ^{0,7}.

The non-identity central elements of the Lie groups are permuted as follows

trial(−1) = e_{12...8}, trial(e_{12...8}) = −e_{12...8} in Spin(8),
trial(−1) = −e_{12...7}, trial(−e_{12...7}) = e_{12...7} in $pin(8).

Exercises Show that

1. For U ∈ G_{2}, U w = −U, Uw = −U, U U = −|U|^{2}, |U ∧ U| = |U|^{2}.

2. For U_{0} ∈ ⋀^{2} ℝ^{8}, U_{1} = Trial(U_{0}), U_{2} = Trial(U_{1}), U_{0} + U_{1} + U_{2} ∈ G_{2}, 2U_{0} − U_{1} − U_{2} ∈ G_{2}^{⊥}.

3. In Cℓ_{0,7}, (1 + w)(1 − e_{12...7}) = (1 − e_{124})(1 − e_{235})(1 − e_{346})(1 − e_{457}), (1 + w)(1 − e_{12...7}) = (1 − v)(1 − e_{12...7}).

4. w^{2} = 7 + 6w.

5. w^{n} = \{ \begin{matrix}\frac{1}{8}(7^{n} − 1) + 1 + \frac{1}{8}(7^{n} − 1)w, n even, \\ \frac{1}{8}(7^{n} + 1) − 1 + \frac{1}{8}(7^{n} + 1)w, n odd. \end{matrix}

19 Note that x F corresponds to ⟨x F̂⟩_{0,1}.

6. For x ∈ ℝ, f(xw) = f(−x)\frac{1}{8}(7 − w) + f(7x)\frac{1}{8}(1 + w). Hint: the minimal polynomial of w, x^{2} = 7 + 6x, has roots x = −1, x = 7.

7. For G ∈ G_{2}, \frac{1}{8}(1 + w)G\frac{1}{8}(1 + w) = 0, \frac{1}{8}(7 − w)G\frac{1}{8}(7 − w) = G, e^{G} = \frac{1}{8}(7 − w)e^{G} \frac{1}{8}(7 − w) + \frac{1}{8}(1 + w).

8. v^{2} = −7 − 6w in Cℓ_{8}, v^{2} = 7 + 6w in Cℓ_{0,7}.

9. For v ∈ ⋀^{3} ℝ^{7}, e^{πv} = −1. Hint: v^{4} + 50v^{2} + 49 = 0, and x^{4} + 50x^{2} + 49 = 0 has roots ±i, ±7i.

10. For v ∈ ⋀^{3} ℝ^{0,7}, cos(πv) = −1, sin(πv) = 0. Hint: v^{4} − 50v^{2} + 49 = 0, and x^{4} − 50x^{2} + 49 = 0 has roots ±7, ±1.

11. For F ∈ ⋀^{2} ℝ^{8}, F = G + H, G ∈ G_{2}, H ∈ G_{2}^{⊥} : H = \frac{1}{3}w (w ∧ F).

12. For H ∈ G_{2}^{⊥}, H = \frac{1}{4}w (w H).

13. For u_{0} ∈ $pin(8), u_{1} = trial(u_{0}), u_{2} = trial(u_{1}) : u_{0}û_{1}^{−1}u_{2} ∈ $pin(7). Hint: a = u_{0}^{−1}û_{0} ∈ ℝ ⊕ ℝ^{0,7} and so a = { \atop trial(a)}trial(a)^{−1}.

14. For the opposite product x • y = y ∘ x of x, y ∈ 𝕆 = ℝ^{8}, ǔ_{0}(x • y)ǔ_{0}^{−1} = (u_{2}xu_{2}^{−1}) • (u_{1}yu_{1}^{−1}).

15. u_{0}(x ∘ y)u_{0}^{−1} = (ǔ_{2}xǔ_{2}^{−1}) ∘ (ǔ_{1}yǔ_{1}^{−1}).

16. ⟨û_{0}x(1 + w)(1 − e_{12...7})⟩_{0,1} = u_{1}xû_{1}^{−1}, ⟨û_{0}x(1 + w)(1 + e_{12...7})⟩_{0,1} = u_{2}xû_{2}^{−1}.

17. ⟨û_{0}xy(1 + w)(1 − e_{12...7})⟩_{0,1} = u_{1}(x ∘ y)û_{1}^{−1}, ⟨û_{0}xy(1 + w)(1 + e_{12...7})⟩_{0,1} = u_{2}(y ∘ x)û_{2}^{−1}.

18. For B ∈ ⋀^{2} ℝ^{8} ∩ Spin(8), ⟨trial(B)⟩_{0} = \frac{1}{4}. For C ∈ ⋀^{4} ℝ^{8} ∩ Spin(8), ⟨trial(C)⟩_{0} = 0. For D ∈ ⋀^{6} ℝ^{8} ∩ Spin(8), ⟨trial(D)⟩_{0} = −\frac{1}{4}.

19. For C ∈ ⋀^{4} ℝ^{8} ∩ Spin(8), trial(C) ∈ ⋀^{4} ℝ^{8}.

20. For u ∈ Spin(8), inducing a simple rotation U = ρ(u) :

⟨trial(u)⟩_{8}e_{12...8} > 0 and ⟨trial(trial(u))⟩_{8}e_{12...8} < 0.

21. ⟨G_{2}⟩_{0} ≥ −\frac{1}{8}, ⟨trial(Spin(7))⟩_{0} ≥ −\frac{1}{4}.

22. G_{2} ∩ exp(G_{2}^{⊥} ∩ ⋀^{2} ℝ^{7}) is homeomorphic but not isometric to S^{6}.

23. diam(G_{2}) = \frac{3}{2}, diam(trial(Spin(7))) = \sqrt{\frac{5}{2}}.

24. Triality does not extend to an automorphism of Pin(8).

25. [\frac{1}{4}(w − 3)]^{2} = 1. Determine

26. The matrix of \frac{1}{4}(w − 3) in the basis (f_{1}, f_{2}, …, f_{8}) of Cℓ_{8}^{+}f, where f_{i} = e_{i}e_{8}f, i = 1, 2, …, 8, and f = \frac{1}{8}(1 + w)\frac{1}{2}(1 + e_{12...8}).

Solutions

13. u_{0}^{−1}û_{0} = û_{1}^{−1}u_{2}û_{2}^{−1}u_{1} so 1 = u_{0}û_{1}^{−1}u_{2}(û_{2}^{−1}u_{1}û_{0}^{−1}) = u_{0}û_{1}^{−1}u_{2}(û_{0}u_{1}^{−1}û_{2})^{−1}

which implies u_{0}û_{1}^{−1}u_{2} = û_{0}u_{1}^{−1}û_{2} = { \atop (u_{0}û_{1}^{−1}u_{2})}.

24. Triality sends −1 ∈ Cen(Pin(8)) to e_{12...8} ∉ Cen(Pin(8)).

23.12 Factorization of u ∈ Spin(8) 319

26.

16⟨f̃_{i} \frac{1}{4}(w − 3)f_{j}⟩_{0} = ( \begin{matrix}−1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) .

Bibliography

J.F. Adams: On the non-existence of elements of Hopf invariant one. Ann. of Math. 72 (1960), 20-104. J.F. Adams: Vector fields of spheres. Ann. of Math. 75 (1962), 603-632. F. van der Blij: History of the octaves. Simon Stevin 34 (1961), 106-125. C. Chevalley: The Algebraic Theory of Spinors. Columbia University Press, New York, 1954. The Algebraic Theory of Spinors and Clifford Algebras. Springer, Berlin, 1997. G.M. Dixon: Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics. Kluwer, Dordrecht, 1994. H.-D. Ebbinghaus et al. (eds.): Numbers. Springer, New York, 1991. A.J. Hahn: Cayley algebras and the automorphisms of P O′_{8}(V) and P Ω′_{8}(V). Amer. J. Math. 98 (1976), 953-987. A.J. Hahn: Cayley algebras and the isomorphisms of the orthogonal groups over arithmetic and local domains. J. Algebra 45 (1977), 210-246. F.R. Harvey: Spinors and Calibrations. Academic Press, San Diego, 1990. M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol: The Book of Involutions. American Mathematical Society, Colloquium Publications 44, 1998. W.S. Massey: Cross products of vectors in higher dimensional Euclidean spaces. Amer. Math. Monthly 90 (1983), #10, 697-701. A. Micali, Ph. Revoy: Modules quadratiques. Cahiers Mathématiques 10, Montpellier, 1977. Bull. Soc. Math. France 63, suppl. (1979), 5-144. S. Okubo: Octonions and Non-associative Algebras in Physics. Cambridge University Press, Cambridge, 1994. I.R. Porteous: Clifford Algebras and the Classical Groups. Cambridge University Press, Cambridge, 1995. R.D. Schafer: On the algebras formed by the Cayley-Dickson process. Amer. J. Math. 76 (1954), 435-446. T.A. Springer, F.D. Veldkamp: Octonions, Jordan Algebras and Exceptional Groups. Springer, Berlin, 2000. G.P. Wene: A construction relating Clifford algebras and Cayley-Dickson algebras. J. Math. Phys. 25 (1984), 2351-2353.

A History of Clifford Algebras

Clifford’s geometric algebras were created by William K. Clifford in 1878/1882, when he introduced a new multiplication rule into Grassmann’s exterior algebra ⋀ ℝ^{n}, by means of an orthonormal basis (e_{1}, e_{2}, …, e_{n}) of ℝ^{n}. Clifford also classified his algebras into four classes according to the signs in (e_{1}e_{2} … e_{n})^{2} = ±1 and (e_{1}e_{2} … e_{n})e_{i} = ±e_{i}(e_{1}e_{2} … e_{n}). In the special case of n = 3, Clifford’s construction embodied Hamilton’s quaternions, as bivectors i = e_{2}e_{3}, j = e_{3}e_{1}, k = e_{1}e_{2}. Clifford algebras were independently rediscovered by Lipschitz 1880/1886, who also presented their first application to geometry, while exploring rotations of ℝ^{n}, in terms of Spin(n), a normalized subgroup of the Lipschitz group Γ_{n}^{+}.

Spinor representations of the orthogonal Lie algebras, B_{ℓ} = so(2n + 1) and D_{ℓ} = so(2n), were observed by E. Cartan in 1913, but without using the term “spinor”. Two-valued spinor representations of the rotation groups SO(n) were re-constructed recursively by Brauer & Weyl in 1935, but without using the term “Clifford algebra”.

In the Schrödinger equation, Pauli 1927 replaced π^{2} = π⃗ · π⃗, where π⃗ = −iℏ∇ − eA⃗, by

π^{2} = π⃗ · π⃗ + π⃗ ∧ π⃗,

where the exterior part does not vanish: (π⃗ ∧ π⃗)ψ = −ℏeB⃗ψ. Pauli explained interaction of a spinning electron with a magnetic field B⃗ by means of a spinorvalued wave function ψ : ℝ^{3} × ℝ → ℂ^{2}. Thus, besides tensors, nature required new kinds of objects: spinors, whose construction calls for Clifford algebras.

Clifford algebras are not only necessary but also offer advantages: their multivector structure enables controlling of subspaces without losing information about their orientations. Physicists are familiar with this advantage in the special case of 1-dimensional oriented subspaces, which they manipulate by vectors, not by projection operators, which forget orientations.

23.13 Algebras of Hamilton, Grassmann and Clifford The first step towards a Clifford algebra was taken by Hamilton in 1843 (first published 1844), when he studied products of sums of squares and invented his quaternions while searching for multiplicative compositions of triplets in ℝ^{3}. The present formulation of vector algebra was extracted from the quaternion product of triplets/vectors xy = −x·y+x×y by Gibbs 1881-84 (first published 1901). Hamilton regarded quaternions as quotients of vectors and wrote a rotation in the form y = ax using a unit quaternion a ∈ ℍ, |a| = 1. In such a rotation, the axis a had to be perpendicular to the vector x (which turned in the plane orthogonal to a). Cayley 1845 published the formula for rotations

y = axa^{−1}, a = cos(\frac{α}{2}) + \frac{a}{α} sin(\frac{α}{2}),

about an arbitrary axis a ∈ ℝ^{3} by angle α = |a| (Cayley ascribed the discovery to Hamilton). Cayley thus came into contact with half-angles and spin representation of rotations in ℝ^{3}. However, in 1840 Olinde Rodrigues had already recognized the relevancy of half-angles in his study on the composition of rotations in ℝ^{3}. Cayley 1855 also discovered the quaternionic representation q → aqb of rotations in ℝ^{4}, equivalent to the decomposition Spin(4) ≃ Spin(3) × Spin(3).

The quaternion algebra ℍ is isomorphic to the even Clifford algebras Cℓ_{3}^{+} ≃ Cℓ_{0,3}^{+} as well as to the proper ideals Cℓ_{0,3} \frac{1}{2}(1 ± e_{123}) of Cℓ_{0,3}. Hamilton also considered complex quaternions ℂ ⊗ ℍ, isomorphic to the Clifford algebra Cℓ_{3} on ℝ^{3}. Note the algebra isomorphisms Cℓ_{3} ≃ Mat(2, ℂ) and Cℓ_{0,3} ≃ ℍ ⊕ ℍ.

Bivectors were introduced by H. Grassmann, when he created his exterior algebras in 1844. The exterior product of two vectors, the bivector a ∧ b, was interpreted geometrically as the parallelogram with a and b as edges, and two such exterior products were equal if their parallelograms lay in parallel planes and had the same area with the same sense of rotation (from a to b). Thus the exterior product of two vectors was anticommutative, a ∧ b = −b ∧ a. Using a basis (e_{1}, e_{2}, …, e_{n}) for ℝ^{n}, the exterior algebra ⋀ ℝ^{n} had a basis

1 e_{1}, e_{2}, …, e_{n} e_{1} ∧ e_{2}, e_{1} ∧ e_{3}, …, e_{1} ∧ e_{n}, e_{2} ∧ e_{3}, …, e_{n−1} ∧ e_{n}

⋮ e_{1} ∧ e_{2} ∧ … ∧ e_{n}

and was thereby of dimension 2^{n}.

W. K. Clifford studied compounds (tensor products) of two quaternion algebras, where quaternions of one algebra commuted with the quaternions of the other algebra, and applied exterior algebra to grade these tensor products of quaternion algebras. Clifford coined his geometric algebra in 1876 (first published in 1878). Clifford’s geometric algebra was generated by n orthogonal unit vectors e_{1}, e_{2}, …, e_{n} which anticommuted e_{i}e_{j} = −e_{j}e_{i} (like Grassmann) and satisfied all e_{i}^{2} = −1 (like Hamilton) [or then all e_{i}^{2} = 1 as in Clifford’s paper 1882]. The number of independent products e_{i}e_{j} = e_{i} ∧ e_{j}, i < j, of degree 2 was \frac{1}{2}n(n − 1) = \binom{n}{2}. Clifford summed up the numbers of independent products of various degrees 0, 1, 2, …, n and thereby determined the dimension of his geometric algebra to be

1 + n + \binom{n}{2} + … + 1 = 2^{n}.

Clifford distinguished four classes of these geometric algebras characterized by the signs of (e_{1}e_{2} … e_{n})e_{i} = ±e_{i}(e_{1}e_{2} … e_{n}) and (e_{1}e_{2} … e_{n})^{2} = ±1. He also introduced two algebras of lower dimension 2^{n−1}, namely, the subalgebra of even elements and, for odd n, a reduced (non-universal) algebra obtained by putting e_{1}e_{2} … e_{n} = ±1 [instead of letting e_{1}e_{2} … e_{n} = e_{1} ∧ e_{2} ∧ … ∧ e_{n} be of degree n with (e_{1}e_{2} … e_{n})^{2} = 1].

23.14 Rudolf Lipschitz Clifford’s geometric algebra was reinvented in 1880, just two years after its first publication, by Rudolf Lipschitz, who later acknowledged Clifford’s prior discovery in his book, see R. Lipschitz: Untersuchungen über die Summen von Quadraten, 1886. In his study on sums of squares, Lipschitz considered a representation of rotations by complex numbers and quaternions and generalized this to higher dimensional rotations in ℝ^{n}. Lipschitz thus gave the first geometric application of Clifford algebras in 1880. He expressed a rotation

y = (I + A)(I − A)^{−1}x, x ∈ ℝ^{n},

(written here in modern notation with an antisymmetric matrix A) in the form y − Ay = x + Ax or as y + y B = x + B x where B ∈ ⋀^{2} ℝ^{n} is the bivector determined by Ax = B x ( = −x B). Lipschitz rewrote (I −A)y = (I +A)x using an even Clifford number a ∈ Cℓ_{0,n}^{+} in the form ya = ax, thus representing the rotation as

y = axa^{−1}, a ∈ Γ_{0,n}^{+}.

[Lipschitz wrote ya_{1} = ax where x = xe_{1}^{−1}, y = ye_{1}^{−1}, a_{1} = e_{1}ae_{1}^{−1}.] In modern terms (introduced by Chevalley), Lipschitz used the exterior exponential

a = e^{∧B}, e^{∧B} = 1 + B + \frac{1}{2}B ∧ B + \frac{1}{6}B^{∧3} + ⋯,

of the bivector B so that the normalized element a/|a| was in the spin group Spin(n).

23.15 Theodor Vahlen Vahlen 1897 found an explicit expression for the multiplication rule of two basis elements in Cℓ_{0,n}

(e_{1}^{α_{1}}e_{2}^{α_{2}} … e_{n}^{α_{n}})(e_{1}^{β_{1}}e_{2}^{β_{2}} … e_{n}^{β_{n}}) = (−1)^{∑^{i≥j} α_{i}β_{j}}e_{1}^{α_{1}+β_{1}}e_{2}^{α_{2}+β_{2}} … e_{n}^{α_{n}+β_{n}}

where the exponents are 0 or 1 (added here modulo 2, although for Vahlen 1+1 = 2, so that summation was over i > j). Vahlen’s formula has frequently been reinvented afterwards: for positive metrics by Brauer & Weyl 1935, for arbitrary metrics by Deheuvels 1981 p. 294, disguised with index sets as in Chevalley 1946 p. 62, Artin 1957 p. 186 and Brackx & Delanghe & Sommen 1982 p. 2, or hidden among permutations as in Kähler 1960/62 and Delanghe & Sommen & Souček 1992, pp. 58-59.

Although Brauer & Weyl 1935 reinvented (in the case of the Clifford algebra Cℓ_{n}) the above explicit multiplication formula of Vahlen, they did not observe the connection to the Walsh functions (discovered in the meantime by Walsh 1923). The connection to the Walsh functions was observed by Hagmark & Lounesto 1986.

In 1902, Vahlen initiated the study of Möbius transformations of vectors in ℝ^{n} (or paravectors in ℝ ⊕ ℝ^{n}) by 2 × 2-matrices with entries in Cℓ_{0,n}. This study was re-initiated by Ahlfors in the 1980’s.

23.16 Elie Cartan Besides detecting spinors in 1913 (and pure spinors in 1938), Cartan made two other contributions to Clifford algebras: their periodicity of 8 and the triality of Spin(8).

Cartan 1908 p. 464, identified the Clifford algebras Cℓ_{p,q} as matrix algebras with entries in ℝ, ℂ, ℍ, ℝ⊕ℝ, ℍ⊕ℍ and found a periodicity of 8. To decipher Cartan’s notation:

h
±1 S_{m} Mat(m, ℝ)
±2 IS_{m} Mat(m, ℂ)
±3 QS_{m} Mat(m, ℍ)
0 2S_{m} ^{2}Mat(m, ℝ)
4 2Q_{m} ^{2}Mat(m, ℍ)

where h = 1 − p + q (mod 8). Clifford’s original notion of 4 classes was thus refined to 8 classes (and generalized from Cℓ_{n} and Cℓ_{0,n} to Cℓ_{p,q}). ^{20}

Cartan’s periodicity of 8 for real Clifford algebras, with an involution, was extended by C.T.C Wall 1968 and Porteous 1969 (rediscovered by Harvey 1990). Wall considered real graded Clifford algebras, with an anti-involution, and found a 2-way periodicity of type (8 × 8)/2, like the movements of a bishop on a chessboard. Porteous used the anti-involution to induce a scalar product for spinors, and classified the scalar products of spinors into 32 classes, according to the signature types (p, q) of real quadratic spaces ℝ^{p,q}.

In 1925, Cartan came into contact with the triality automorphism of Spin(8). Lounesto 1997 (in the first edition of this book) showed that triality is a restriction of a polynomial mapping Cℓ_{8} → Cℓ_{8}, of degree 2.

23.17 Ernst Witt Witt 1937 started the modern algebraic theory of quadratic forms. He identified Clifford algebras of non-degenerate quadratic forms over arbitrary fields of characteristic ≠ 2. The Witt ring W (𝔽), of a field 𝔽, consists of similarity classes of non-singular quadratic forms over 𝔽 (similar quadratic forms have isometric anisotropic parts). In characteristic ≠ 2, the structure of Clifford algebras of certain quadratic forms was studied by Lee 1945/48 (e_{i}^{2} = 1), Chevalley 1946 (e_{i}^{2} = −1), and Kawada & Iwahori 1950 (e_{i}^{2} = ±1). These authors did not benefit the Witt ring (although they already had it at their disposal), and so they did not consider all the isometry classes of anisotropic quadratic forms.

Example. The Witt ring W (𝔽_{5}) of the finite field 𝔽_{5} = {0, 1, 2, 3, 4} of characteristic 5 contains four isometry classes 0, ⟨1⟩, ⟨s⟩, ⟨1, s⟩ where s = 2 or s = 3. Chevalley 1946, Lee and Kawada & Iwahori did not notice that none of the quadratic forms x_{1}^{2} + x_{2}^{2}, x_{1}^{2} − x_{2}^{2}, −x_{1}^{2} − x_{2}^{2} on the plane 𝔽_{5}^{2} is isometric with x_{1}^{2} + sx_{2}^{2} ≃ ⟨1, s⟩ (but in fact they are all neutral, and thereby in the same isometry class as 0). A simpler example is the line 𝔽_{5} where the Clifford algebra of 2x^{2} ≃ ⟨2⟩ is the quadratic extension 𝔽_{5}(\sqrt{2}) whereas the Clifford algebras of both ±x^{2} ≃ ⟨±1⟩ split 𝔽_{5} × 𝔽_{5} .

20 Cartan’s periodicity of 8 for Clifford algebras, from 1908, is often attributed to Bott, who

was born in 1923 and proved his periodicity of homotopy groups of rotation groups in

23.18 Claude Chevalley Chevalley 1954 constructed Clifford algebras as subalgebras of the endomorphism algebra of the exterior algebra, Cℓ(Q) ⊂ End(⋀ V), by means of a not necessarily symmetric bilinear form B on V such that Q(x) = B(x, x). By this construction, Chevalley managed to include the exceptional case of characteristic 2, and thus amended the work of Witt.

Chevalley 1954 went on further and gave the most general definition, as a factor algebra of the tensor algebra, Cℓ(Q) = ⊗V/I_{Q}, valid also when ground fields are replaced by commutative rings. From the pedagogical point of view, this approach is forbidding, while it refers to the infinite-dimensional tensor algebra ⊗V.

Chevalley 1954 introduced exterior exponentials of bivectors and used them to scrutinize the Lipschitz group, unfairly naming it a ‘Clifford group’. Thus, there were two exponentials such that

e^{B} ∈ Spin(n) and \frac{e^{∧B}}{|e^{∧B}|} ∈ Spin(n) for B ∈ ⋀^{2} ℝ^{n}.

23.19 Marcel Riesz Marcel Riesz 1958, pp. 61-67, reconstructed Grassmann’s exterior algebra from the Clifford algebra, in any characteristic ≠ 2, by

x ∧ u = \frac{1}{2}(xu + (−1)^{k}ux),

where x ∈ V and u ∈ ⋀^{k} V. Riesz’ contribution enhanced Chevalley’s result of 1946, which related exterior products of vectors to antisymmetrized Clifford products of vectors (Chevalley’s result was valid only in characteristic 0).

Clifford algebras admit a parity grading or even-odd grading with even and odd parts, Cℓ(Q) = Cℓ^{+}(Q)⊕Cℓ^{−}(Q). Riesz’s construction of 1958 showed that there also exists a dimension grading

Cℓ(Q) = 𝕂 ⊕ V ⊕ ⋀^{2} V ⊕ … ⊕ ⋀^{n} V

in all characteristics ≠ 2 (because there is a privileged linear isomorphism between a Clifford algebra and the exterior algebra). Thus, bivectors exist in all characteristics ≠ 2.

M. Riesz 1947 expressed the Maxwell stress tensor as T_{μν} = −\frac{1}{2}⟨e_{μ}Fe_{ν}F⟩_{0}. The first one to consider spinors as elements in a minimal left ideal of a

Clifford algebra was M. Riesz 1947 (although the special case of pure spinors had been considered earlier by Cartan 1938).

23.20 Atiyah & Bott & Shapiro In 1964, Atiyah & Bott & Shapiro reconsidered spinor spaces as modules over a Clifford algebra, instead of regarding them as minimal left ideals in the Clifford algebra. This permitted differentiation of spinor valued functions on manifolds, not just on flat spaces.

They re-identified the definite real Clifford algebras Cℓ_{n} and Cℓ_{0,n} as matrix algebras with entries in ℝ, ℂ, ℍ, ℝ ⊕ ℝ, ℍ ⊕ ℍ (identified by Cartan in 1908). They rediscovered the periodicity of 8 (found by Cartan in 1908) with respect to the graded tensor product (used earlier by Chevalley in 1955). They emphasized the importance of the ℤ_{2} -graded structure (the even-odd parity in Clifford’s works) and used it to simplify Chevalley’s approach to Lipschitz groups (the role of parity grading and/or grade involution in reflections was not observed by Chevalley).

Atiyah & Bott & Shapiro’s paper of 1964 is known for its dirty joke: they attributed the introduction of Pin(n) to Serre from France, where pronunciation of ‘pin group’ in English sounds the same as ‘pine groupe’ in French: pine [pin] is a slang expression for male genitals.

E. Kähler 1960/62 introduced a second product for Cartan’s exterior differential forms making Grassmann’s exterior algebra isomorphic with a Clifford algebra. This re-interpretation of Chevalley’s definition of the Clifford algebra (extending xu = x u + x ∧ u) was renamed as Kähler–Atiyah algebra by W. Graf 1978 (Graf’s Kähler–Atiyah algebra was again reinvented and applied to the Kähler–Dirac equation by Salingaros & Wene 1985).

23.21 The Maxwell Equations In the special case of a homogeneous and isotropic medium, Maxwell equations can be condensed into a single equation. This has been done by means of complex vectors (Silberstein 1907), complex quaternions (Silberstein 1912/1914), spinors (Laporte & Uhlenbeck 1931) and in terms of Clifford algebras. Juvet & Schidlof 1932, Mercier 1935 and Riesz 1958 condensed the Maxwell equations into a single equation by bivectors in the Clifford algebra Cℓ_{1,3} of ℝ^{1,3}. In the Clifford algebra Cℓ_{3,1} ≃ Mat(4, ℝ), the single Maxwell equation

∂F = J,

where ∂ = ∇ − e_{0}∂_{0} and F = Ee_{0} − Be_{123} ∈ ⋀^{2} ℝ^{3,1}, could be decomposed into two parts

∂ F = J, ∂ ∧ F = 0.

Similarly, ∂A = −F could be decomposed into two parts, ∂ ∧A = −F and the Lorenz gauge/condition ∂ · A = 0 discovered by the Danish physicist Ludwig Lorenz and not by the Dutch physicist H. A. Lorentz. ^{21} Marcel Riesz 1947 expressed the Maxwell stress tensor as

T_{μν} = −\frac{1}{2}⟨e_{μ}Fe_{ν}F⟩_{0}

and Hestenes 1966, p. 31, introduced the vectors T_{μ} = −\frac{1}{2}Fe_{μ}F for which T_{μν} = T_{μ}·e_{ν} = e_{μ}·T_{ν} and the mapping T x = −\frac{1}{2}FxF where (T x)^{μ} = T^{μ}_{ν}x^{ν}. [Juvet & Schidlof 1932, p. 141, gave

T_{μν} = F_{μ}^{λ}F_{λν} + \frac{1}{4}g_{μν}F_{αβ}F^{αβ}

but did not observe that T x = −\frac{1}{2}FxF.] Note that T_{0} = \frac{1}{2}(E^{2}+B^{2})e_{0}+E×B.

23.22 Spinors in ideals, as operators and recovered Juvet 1930 and Sauter 1930 replaced column spinors by square matrices in which only the first column was non-zero – thus spinor spaces became minimal left ideals in a matrix algebra. Marcel Riesz 1947 was the first one to consider spinors as elements in a minimal left ideal of a Clifford algebra (although the special case of pure spinors had been considered earlier by Cartan 1938).

Gürsey 1956/58 rewrote the Dirac equation with 2 × 2 quaternion matrices in Mat(2, ℍ) (see also Gsponer & Hurni 1993). Kustaanheimo 1964 presented the spinor regularization of the Kepler motion, the KS -transformation, which emphasized the operator aspect of spinors. This led David Hestenes 1966-74 to a reformulation of the Dirac theory, where the role of spinors [in columns ℂ^{4} or in minimal left ideals of the complex Clifford algebra ℂ ⊗ Cℓ_{1,3} ≃ Mat(4, ℂ)] was taken over by operators in the even subalgebra Cℓ_{1,3}^{+} of the real Clifford algebra Cℓ_{1,3} ≃ Mat(2, ℍ).

Spinors were reconstructed from their bilinear covariants by Y. Takahashi 1983 and J. Crawford 1985, in the case of the electron. Lounesto 1993 generalized the reconstruction of spinors to the null case of the neutron, and predicted existence of a new particle residing in between electrons and neutrons.

Bibliography

E. Artin: Geometric Algebra. Interscience, New York, 1957, 1988.

21 J. van Bladel: Lorenz or Lorentz? IEEE Antennas and Propagation Magazine 33 (1991)

p. 69 and The Radioscientist 2 (1991) p. 55.

M.F. Atiyah, R. Bott, A. Shapiro: Clifford modules. Topology 3, suppl. 1 (1964), 3-38. Reprinted in R. Bott: Lectures on K(X). Benjamin, New York, 1969, pp. 143-178. Reprinted in Michael Atiyah: Collected Works, Vol. 2. Clarendon Press, Oxford, 1988, pp. 301-336. R. Brauer, H. Weyl: Spinors in n dimensions. Amer. J. Math. 57 (1935), 425-449. Reprinted in Selecta Hermann Weyl, Birkhäuser, Basel, 1956, pp. 431-454. E. Cartan (exposé d’après l’article allemand de E. Study): Nombres complexes, pp. 329-468 in J. Molk (red.): Encyclopédie des sciences mathématiques, Tome I, vol. 1, Fasc. 4, art. I5, 1908. E. Cartan: Sur les groupes projectifs qui ne laissent invariante aucune multiplicité plane. Bull. Soc. Math. France 41 (1913), 53-96. E. Cartan: Le principe de dualité et la théorie des groupes simples et semi-simples. Bull. Sci. Math. 49 (1925), 161-174. A. Cayley: On certain results relating to quaternions. Phil. Mag. (3) 26 (1845), 141-145. A. Cayley: Recherches ultérieures sur les déterminants gauches. J. Reine Angew. Math. 50 (1855), 299-313. C. Chevalley: Theory of Lie Groups. Princeton University Press, Princeton, 1946. C. Chevalley: The Algebraic Theory of Spinors. Columbia University Press, New York, 1954. The Algebraic Theory of Spinors and Clifford Algebras. Springer, Berlin, 1997. C. Chevalley: The Construction and Study of Certain Important Algebras. Mathematical Society of Japan, Tokyo, 1955. W.K. Clifford: Applications of Grassmann’s extensive algebra. Amer. J. Math. 1 (1878), 350-358. W.K. Clifford: On the classification of geometric algebras; pp. 397-401 in R. Tucker (ed.): Mathematical Papers by William Kingdon Clifford. Macmillan, London, 1882. (Reprinted by Chelsea, New York, 1968.) Title of talk announced already on p. 135 in Proc. London Math. Soc. 7 (1876). J.P. Crawford: On the algebra of Dirac bispinor densities: Factorization and inversion theorems. J. Math. Phys. 26 (1985), 1439-1441. P.A.M. Dirac: The quantum theory of the electron. Proc. Roy. Soc. A 117 (1928), 610-624. A. Gsponer, J.-P. Hurni: Lanczos’ equation to replace Dirac’s equation? in pp. 509-512 of J.D. Brown (ed.): Proceedings of the Cornelius Lanczos Centenary Conference, (Raleigh, NC, 1993). F. Gürsey: Correspondence between quaternions and four-spinors. Rev. Fac. Sci. Univ. Istanbul A21 (1956), 33-54. F. Gürsey: Relation of charge independence and baryon conservation to Pauli’s transformation. Nuovo Cimento 7 (1958), 411-415. P.-E. Hagmark, P. Lounesto: Walsh functions, Clifford algebras and Cayley-Dickson process, pp. 531-540 in J.S.R. Chisholm, A.K. Common (eds.): Clifford Algebras and their Applications in Mathematical Physics. Reidel, Dordrecht, The Netherlands, 1986. J. Helmstetter: Algèbres de Clifford et algèbres de Weyl. Cahiers Math. 25, Montpellier, 1982. D. Hestenes: Space-Time Algebra. Gordon and Breach, New York, 1966, 1987, 1992. D. Hestenes: Real spinor fields. J. Math. Phys. 8 (1967), 798-808. D. Hestenes, G. Sobczyk: Clifford Algebra to Geometric Calculus. Reidel, Dordrecht, 1984, 1987.

23.22 Spinors in ideals, as operators and recovered 329

B. Jancewicz: Multivectors and Clifford Algebra in Electrodynamics. World Scientific Publ., Singapore, 1988. G. Juvet: Opérateurs de Dirac et équations de Maxwell. Comment. Math. Helv. 2 (1930), 225-235. G. Juvet, A. Schidlof: Sur les nombres hypercomplexes de Clifford et leurs applications à l’analyse vectorielle ordinaire, à l’électromagnetisme de Minkowski et à la théorie de Dirac. Bull. Soc. Neuchat. Sci. Nat., 57 (1932), 127-147. E. Kähler: Der innere Differentialkalkül. Rendiconti di Matematica e delle sue Applicazioni (Roma) 21 (1962), 425-523. Y. Kawada, N. Iwahori: On the structure and representations of Clifford algebras. J. Math. Soc. Japan 2 (1950), 34-43. P. Kustaanheimo, E. Stiefel: Perturbation theory of Kepler motion based on spinor regularization. J. Reine Angew. Math. 218 (1965), 204-219. O. Laporte, G. E. Uhlenbeck: Application of spinor analysis to the Maxwell and Dirac equations. Phys. Rev. 37 (1931), 1380-1397. H. C. Lee: On Clifford algebras and their representations. Ann. of Math. 49 (1948), 760-773. R. Lipschitz: Principes d’un calcul algébrique qui contient comme espèces particulières le calcul des quantités imaginaires et des quaternions. C.R. Acad. Sci. Paris 91 (1880), 619-621, 660-664. Reprinted in Bull. Soc. Math. (2) 11 (1887), 115-120. R. Lipschitz: Untersuchungen über die Summen von Quadraten. Max Cohen und Sohn, Bonn, 1886, pp. 1-147. (The first chapter of pp. 5-57 translated into French by J. Molk: Recherches sur la transformation, par des substitutions réelles, d’une somme de deux ou troix carrés en elle-même. J. Math. Pures Appl. (4) 2 (1886), 373-439. French résumé of all three chapters in Bull. Sci. Math. (2) 10 (1886), 163-183.) R. Lipschitz (signed): Correspondence. Ann. of Math. 69 (1959), 247-251. Reprinted on pages 557-561 of A. Weil: Œuvres Scientifiques, Collected Papers, Volume II. Springer, 1979. P. Lounesto: Clifford algebras and Hestenes spinors. Found. Phys. 23 (1993), 1203-1237. A. Mercier: Expression des Équation de l’Électromagnetisme au Moyen des Nombres de Clifford. Thèse, Université de Genéve, 1935. W. Pauli: Zur Quantenmechanik des magnetischen Elektrons. Z. Phys. 42 (1927), 601-623. I.R. Porteous: Topological Geometry. Van Nostrand Reinhold, London, 1969. Cambridge University Press, Cambridge, 1981. I.R. Porteous: Clifford Algebras and the Classical Groups. Cambridge University Press, Cambridge, 1995. M. Riesz: Sur certain notions fondamentales en théorie quantique relativiste. C. R. 10^{e} Congrès Math. Scandinaves, (Copenhagen, 1946). Jul. Gjellerups Forlag, Copenhagen, 1947, pp. 123-148. Reprinted in L. Gårding, L. Hörmander (eds.): Marcel Riesz, Collected Papers. Springer, Berlin, 1988, pp. 545-570. M. Riesz: Clifford Numbers and Spinors. University of Maryland, 1958. Reprinted as facsimile by Kluwer, Dordrecht, 1993. F. Sauter: Lösung der Diracschen Gleichungen ohne Spezialisierung der Diracschen Operatoren. Z. Phys. 63 (1930), 803-814. Y. Takahashi: A passage between spinors and tensors. J. Math. Phys. 24 (1983), 1783-1790. K. Th. Vahlen: Über höhere komplexe Zahlen. Schriften der phys.-ökon. Gesellschaft zu Königsberg 38 (1897), 72-78.

K. Th. Vahlen: Über Bewegungen und complexe Zahlen. Math. Ann. 55 (1902), 585-593. C.T.C. Wall: Graded algebras, antiinvolutions, simple groups and symmetric spaces. Bull. Amer. Math. Soc. 74 (1968), 198-202. B.L. van der Waerden: On Clifford algebras. Nederl. Akad. Wetensch. Proc. Ser. A 69 (1966), 78-83. E. Witt: Theorie der quadratischen Formen in beliebigen Körpern. J. Reine Angew. Math. 176 (1937), 31-44.