Computing Chevalley bases in small characteristics

TL;DR

This paper develops new methods, implemented in Magma, for computing Chevalley bases of simple Lie algebras over fields of small characteristics, specifically 2 and 3, where previous techniques were insufficient.

Contribution

It introduces novel algorithms for computing Chevalley bases in small characteristics, extending existing methods to cases characteristic 2 and 3.

Findings

New algorithms for characteristic 2 and 3 cases

Implementation of methods in Magma

Successful computation of Chevalley bases in challenging cases

Abstract

Let L be the Lie algebra of a simple algebraic group defined over a field F and let H be a split Cartan subalgebra of L. Then L has a Chevalley basis with respect to H. If the characteristic of F is not 2 or 3, it is known how to find it. In this paper, we treat the remaining two characteristics. To this end, we present a few new methods, implemented in Magma, which vary from the computation of centralisers of one root space in another to the computation of a specific part of the Lie algebra of derivations of $L$.