Simple Lie algebras of small characteristic IV.: Solvable and classical roots

Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic p>3 and T a torus of maximal dimension in the p-envelope of L in DerL. In this paper we describe the T-semisimple quotients of the 2-sections of L relative to T and prove that if all 1-sections of L relative to T are compositionally classical or solvable then L is either classical or a Block algebra or a filtered Lie algebra of type S.