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 $\mathrm{Gal}({\mathbb{F}}_8/{\mathbb{F}}_2)$ act on the roots as $120$-degree rotations and complex conjugation acts as negation.

6 pages, 2 figures, submitted

References in corpus (1)

• The octonions as a twisted group algebra