Abstract
Let X(a,b,c) be a ternary vector cross product for eight-dimensional Euclidean space E. An identity is derived which expresses (X(a,b,c), X(u,v,w)) in terms of the Spin(7)-invariant scalar quadruple product Phi (a,b,c,d)=(a,X(b,c,d)). The proof of the identity is coordinate free, and starts out from an explicit expression for X, with E viewed as complex four-dimensional Hilbert space.
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