On the toral rank of simple Lie algebra over a field of characteristic 2 Alexander Grishkov, University of Sao Paulo, Brazil e-mail: grishkov@ime.usp.br 1 Abstract We give a short proof of the following result of S.Skryabin [1]: a finite di- mensional Lie algebra over a field of characteristic 2 and absolute toral rank one is solvable. 2 Introduction. In this note we give a short proof of the following theorem of S.Skryabin [1]. Theorem 1 Finite dimensional simple Lie algebra over a field of character- istic 2 has absolute toral rank at least 2. Proof. Let F be an algebraically closed field of characteristic 2, let ˜ L be a finite dimensional simple Lie algebra over F of toral rank one, let L be a 2-envelope of ˜ L. Then L = L 0 ⊕ L 1 , where L 0 is a Cartan subalgebra of L, [L 0 ,L 1 ] ⊆ L 1 and [L 1 ,L 1 ] ⊆ L 0 . By the definition of absolute toral rank, L 0 contains a unique toroidal element h = h [2] . We denote N = {a ∈ L|∃n : a [2 n ] =0}, T = {t ∈ L|t [2] = t =0}. We need the following simple result. Lemma 1 Let n ∈N , t, s ∈T . For any a ∈ L, if a +[a, n]=0 then a =0. If [t, s]=0 then t = s. 1