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Let $k$ be a nonperfect field of characteristic $2$. Let $G$ be a $k$-splitsimple algebraic group of type $E_6$ (or $G_2$) defined over $k$. In thispaper, we present the first examples of nonabelian non-$G$-completely reducible$k$-subgroups of $G$ which are $G$-completely reducible over $k$. Ourconstruction is based on that of subgroups of $G$ acting non-separably on theunipotent radical of a proper parabolic subgroup of $G$ in our previous work.We also present examples with the same property for a non-connected reductivegroup $G$. Along the way, several general results concerning completereducibility over nonperfect fields are proved using the recently proved Titscenter conjecture for spherical buildings. In particular, we show that undermild conditions a $k$-subgroup of $G$ is pseudo-reductive if it is$G$-completely reducible over $k$.

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 1