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

. 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 sim- ple 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 ).