Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields IV: An F4 example

Let k be a nonperfect separably closed field. Let G be a connected reductive algebraic group defined over k. We study rationality problems for Serre's notion of complete reducibility of subgroups of G. In particular, we present the first example of a connected nonabelian k-subgroup H of G that is G-completely reducible but not G-completely reducible over k, and the first example of a connected nonabelian k-subgroup H′ of G that is G-completely reducible over k but not G-completely reducible. This is new: all previously known such examples are for finite (or non-connected) H and H′ only.