Separability and complete reducibility of subgroups of the Weyl group of a simple algebraic group of type E7

Let G be a connected reductive algebraic group defined over an algebraically closed field k. The aim of this paper is to present a method to find triples (G, M, H) with the following three properties. Property 1: G is simple and k has characteristic 2. Property 2: H and M are closed reductive subgroups of G such that H < M < G, and (G, M) is a reductive pair. Property 3: H is G-completely reducible, but not M-completely reducible. We exhibit our method by presenting a new example of such a triple in G = E-7 Then we consider a rationality problem and a problem concerning conjugacy classes as important applications of our construction. (C) 2014 Elsevier Inc. All rights reserved.