# Classical simple Lie 2-algebras of toral rank 3 and a contragredient Lie 2-algebra of toral rank 4

## TL;DR

This paper classifies certain simple Lie 2-algebras based on their toral rank, proving the non-existence of classical types with odd rank and analyzing a specific contragredient example with rank 4.

## Contribution

It establishes the non-existence of classical simple Lie 2-algebras with odd toral rank and provides a detailed analysis of a particular contragredient Lie 2-algebra of rank 4.

## Key findings

- No classical simple Lie 2-algebras with odd toral rank exist.
- The contragredient Lie 2-algebra G(F_{4, a}) has toral rank 4.
- Cartan decomposition of G(F_{4, a}) is provided.

## Abstract

In this paper we show there are no classical type simple Lie 2-algebras with toral rank odd and we also show that the simple contragredient Lie 2-algebra $G(F_{4, a})$ of dimension 34 has toral rank 4, and we give the Cartan decomposition of $G(F_{4, a})$.