# On simple Lie 2-algebra of toral rank 3

## TL;DR

This paper investigates the structure of simple Lie 2-algebras over fields of characteristic 2, establishing non-existence results for certain ranks and dimensions, thus advancing the classification in this characteristic.

## Contribution

It proves that no simple Lie 2-algebras of toral rank 3 exist over algebraically closed fields of characteristic 2 with dimension ≤16.

## Key findings

- No simple Lie 2-algebras with toral rank 3 and dimension ≤16 exist in characteristic 2.
- Simple Lie algebras over fields of characteristic 0 or >3 are classified, but characteristic 2 remains partially unresolved.
- The minimal toral rank for simple Lie algebras in characteristic 2 is at least 2.

## Abstract

Simple Lie algebras of finite dimension over an algebraically closed field of characteristic 0 or $p> 3$ were recently classified. However, the problem over an algebraically closed field of characteristics 2 or 3 there exist only partial results. The first result on the problem of classification of simple Lie algebra of finite dimension over an algebraically closed field of characteristic 2 is that these algebras have absolute toral rank greater than or equal to 2. In this paper we show that there are not simple Lie 2-algebras with toral rank 3 over an algebraically closed field of characteristic 2 and dimension less or equal to 16.