Classical simple Lie $2$-algebras of odd toral rank and a contragredient Lie $2$-algebra of toral rank 4

Abstract

 After the classification of simple Lie algebras over a field of characteristic $p > 3$, the main problem not yet solved in the theory of finite dimensional Lie algebras is the classification of simple Lie algebras over a field of characteristic 2. The first result for this classification problem ensures that all finite dimensional Lie algebras of absolute toral rank 1 over an algebraically closed field of characteristic 2 are soluble. Describing simple Lie algebras (respectively, Lie $2$-algebras) of finite dimension of absolute toral rank (respectively, toral rank) 3 over an algebraically closed field of characteristic 2 is still an open problem. In this paper we show that there are no classical type simple Lie $2$-algebras with toral rank odd and furthermore that the simple contragredient Lie $2$-algebra $G(F_{4, a})$ of dimension 34 has toral rank 4. Additionally, we give the Cartan decomposition of $G(F_{4, a})$.