Explicit Multiplication Tables of Cayley-Dickson Algebras

Abstract

A long-standing unresolved issue in the Cayley-Dickson algebra of dimension 2 n is that there lacks of an explicit multiplication table although it can be constructed via inductive construction. In this article we solve this open problem. We show that the Cayley-Dickson algebra is a twisted group algebra with an explicit twist function σ(A, B) such that e A e B = (−1) σ(A,B) e A⊕B .The same approach also yields similar results for split Cayley-Dickson algebras.