Simple Lie Algebras of Small Characteristic II. Exceptional Roots

LetLbe a finite dimensional simple Lie algebra of absolute toral rank 2 over an algebraically closed field of characteristicp > 3 andTa 2-dimensional torus in the semisimplep-envelope ofL. Suppose thatLis not isomorphic to a Melikian algebra. It is proved in this paper that, for every root α Γ(L, T), the subalgebraK′(α) generated by ∑i *pKiα(whereKiα = {x Liα α([x, L − iα]) = 0}) acts triangulably onL. In particular, this implies that, in the terminology of R. E. Block and R. L. Wilson (1988,J. Algebra114, 115–259), all roots of Γ(L, T) are nonexceptional.