Throughout this paper, let G be a simple algebraic group of exceptional type over an algebraically closed field K of characteristic p. The maximal closed connected subgroups of these groups were determined in [Se], subject to some mild restrictions on the characteristic p. In this article we describe some results from [LS2] concerning arbitrary closed connected reductive subgroups of G, again assuming mild characteristic restrictions (in particular, p = 0 or p > 7 covers all the restrictions).

Before giving detailed statements, we give a general description of the results. Theorems 5 and 6 below determine the embeddings of arbitrary closed connected semisimple subgroups in G: if X is such a subgroup, then X 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. Subsystem subgroups are constructed naturally from subsystems of the root system of G; this therefore determines the embedding of X in G. As a consequence, when p = 0 there are only finitely many conjugacy classes of such subgroups X, whereas there are infinitely many when p > 0.

The proofs are 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. This result can also be used to prove that CG(X) is always reductive (Theorem 2).