Simple Lie algebras of small characteristic II. Exceptional roots

Let L be a finite dimensional simple Lie algebra of absolute toral rank 2 o ver an algebraically closed field of characteristic p > 3 and T a 2-dimensi onal torus in the semisimple p-envelope of L. Suppose that L is not isomorp hic to a Melikian algebra. It is proved in this paper that, for every root alpha is an element of Gamma(L, T), the subalgebra K'(alpha) generated by S igma(i is an element of Fp*)K(i alpha) (where K-i alpha = {x is an element of L-i alpha \ alpha([x, L-i alpha]) = 0}) acts triangulably on L. In parti cular, this implies that, in the terminology of R. E. Block and R. L. Wilso n (1988, J. Algebra 114, 115-259), all roots of Gamma(L,T) are nonexception al. (C) 1999 Academic Press.