An alternate Cayley-Dickson product

TL;DR

This paper reveals a new, highly periodic form of the Cayley-Dickson twist by using an alternative doubling product and basis numbering, leading to a simple formula for basis vector multiplication.

Contribution

It introduces an alternative Cayley-Dickson product and basis numbering that simplifies the twist and provides a closed-form expression for basis vector multiplication.

Findings

A new periodic structure in the Cayley-Dickson twist

A simple closed-form formula for basis vector products

Enhanced understanding of Cayley-Dickson algebra structure

Abstract

Although the Cayley-Dickson algebras are twisted group algebras, little attention has been paid to the nature of the Cayley-Dickson twist. One reason is that the twist appears to be highly chaotic and there are other interesting things about the algebras to focus attention upon. However, if one uses a doubling product for the algebras different from yet equivalent to the ones commonly used and if one uses a numbering of the basis vectors different from the standard basis a quite beautiful and highly periodic twist emerges. This leads easily to a simple closed form formula for the product of any two basis vectors of a Cayley-Dickson algebra.