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

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 $G(F_{4, a})$.