Simple Lie algebras of small characteristic V. The non-Melikian case

Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic p>3. We prove in this paper that if for every torus T of maximal dimension in the p-envelope of adL in DerL the centralizer of T in adL acts triangulably on L, then L is either classical or of Cartan type. As a consequence we obtain that any finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p>5 is either classical or of Cartan type. This settles the last remaining case of the generalized Kostrikin–Shafarevich conjecture (the case where p=7).