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

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.