Simple Lie Algebras having Extremal Elements

TL;DR

This paper proves that most simple finite-dimensional Lie algebras over fields with characteristic not 2 or 3 are generated by extremal elements, simplifying previous complex proofs and filling a gap in the classification.

Contribution

It establishes that, with one exception, simple Lie algebras are generated by extremal elements, providing a more straightforward proof and completing the geometric classification gap.

Findings

Most simple Lie algebras are generated by extremal elements.

The proof simplifies previous complex arguments.

Fills a gap in the geometric classification of Lie algebras.

Abstract

Let L be a simple finite-dimensional Lie algebra of characteristic distinct from 2 and from 3. Suppose that L contains an extremal element that is not a sandwich, that is, an element x such that [x, [x, L]] is equal to the linear span of x in L. In this paper we prove that, with a single exception, L is generated by extremal elements. The result is known, at least for most characteristics, but the proofs in the literature are involved. The current proof closes a gap in a geometric proof that every simple Lie algebra containing no sandwiches (that is, ad-nilpotent elements of order 2) is in fact of classical type.