  Let $k$ be a nonperfect field of characteristic $2$. Let $G$ be a $k$-split
simple algebraic group of type $E_6$ (or $G_2$) defined over $k$. In this
paper, we present the first examples of nonabelian non-$G$-completely reducible
$k$-subgroups of $G$ which are $G$-completely reducible over $k$. Our
construction is based on that of subgroups of $G$ acting non-separably on the
unipotent radical of a proper parabolic subgroup of $G$ in our previous work.
We also present examples with the same property for a non-connected reductive
group $G$. Along the way, several general results concerning complete
reducibility over nonperfect fields are proved using the recently proved Tits
center conjecture for spherical buildings. In particular, we show that under
mild conditions a $k$-subgroup of $G$ is pseudo-reductive if it is
$G$-completely reducible over $k$.
