  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 a new example of subgroup $H$ of $G$ of type $D_4$ in characteristic
$2$ such that $H$ is $G$-completely reducible but not $G$-completely reducible
over $k$ (or vice versa). This is new: all known such examples are for $G$ of
exceptional type. We also find a new counterexample for K\"ulshammer's question
on representations of finite groups for $G$ of type $D_4$. A problem concerning
the number of conjugacy classes is also considered. The notion of nonseparable
subgroups plays a crucial role in all our constructions.
