  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.
