Reductive subgroups of exceptional algebraic groups

Let G be a simple algebraic group of exceptional type over an algebraically closed field of characteristic p. We determine the embeddings of arbitrary closed connected semisimple subgroups in G, under some mild restrictions on p: every such subgroup is embedded in an explicit way in a ''subsystem subgroup'' of G (that is, a semisimple subgroup which is normalized by a maximal torus of G). The proof is based on Theorem 1, which states that if the reductive subgroup X lies in a parabolic subgroup P = QL of G, with unipotent radical Q and Levi subgroup L, then some conjugate of X lies in L. We also use Theorem 1 to prove that CG(X) is always reductive. Other results in the paper concern the actions of simple closed connected subgroups X of G on the Lie algebra L(G) of G. Following Dynkin, a labelled diagram is associated with each such subgroup. We prove that when X is of type Al the labelled diagram determines X up to conjugacy in Aut G; and we establish a similar, though weaker, result when X has rank 2 or more. Further results give information about centralizers and composition factors of X acting on L(G).