Octonion X,Y-Product $G_{2}$ Variants

TL;DR

This paper explores how the automorphism group G2 of octonions varies with different X,Y-product variants, providing a general method to transition between G2 and its variants.

Contribution

It introduces a general solution for deriving G2 variants from the standard automorphism group using octonion X,Y-products.

Findings

Derived a method to obtain G2 variants from the standard automorphism group.

Established a framework for understanding automorphism group changes with X,Y-products.

Provided explicit formulas for transitioning between G2 and its variants.

Abstract

The automorphism group $G_{2}$ of the octonions changes when octonion X,Y-product variants are used. I present here a general solution for how to go from $G_{2}$ to its X,Y-product variant.