Some Graded Identities of The Cayley-Dickson Algebra

TL;DR

This paper determines bases of graded identities for octonion algebras under specific gradings, expanding understanding of their algebraic structure in different characteristic fields.

Contribution

It provides explicit bases of graded identities for octonion algebras with b2^2 and b2^3 gradings, derived from the Cayley-Dickson process, in characteristic not two.

Findings

Identifies graded identities for b2^2 and b2^3 gradings

Establishes basis of identities for octonion algebra in these gradings

Extends previous work on Cayley-Dickson algebras in different characteristics

Abstract

We work to find a basis of graded identities for the octonion algebra. We do so for the $\mathbb{Z}_2^2$ and $\mathbb{Z}_2^3$ gradings, both of them derived of the Cayley-Dickson process, the later grading being possible only when the characteristic of the scalars is not two.