Root space decomposition of $\mathfrak{g}_2$ from octonions

TL;DR

The paper presents a straightforward method to explicitly construct derivations of octonions forming a Chevalley basis of the Lie algebra g_2, utilizing the structure of octonions as a twisted group algebra over _8.

Contribution

It introduces a simple approach to derive g_2 from octonions using their description as a twisted group algebra of _8, highlighting the action of Galois automorphisms.

Findings

Explicit derivations of octonions as a Chevalley basis of g_2

Representation of Galois automorphisms as root rotations

Connection between complex conjugation and root negation

Abstract

We describe a simple way to write down explicit derivations of octonions that form a Chevalley basis of $\mathfrak{g}_2$. This uses the description of octonions as a twisted group algebra of the finite field $\mathbb{F}_8$. Generators of $\operatorname{Gal}(\mathbb{F}_8/\mathbb{F}_2)$ act on the roots as $120$-degree rotations and complex conjugation acts as negation.