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Let $k$ be a nonperfect separably closed field. Let $G$ be a connectedreductive algebraic group defined over $k$. We study rationality problems forSerre's notion of complete reducibility of subgroups of $G$. In particular, wepresent 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 reducibleover $k$ (or vice versa). This is new: all known such examples are for $G$ ofexceptional type. We also find a new counterexample for K\"ulshammer's questionon representations of finite groups for $G$ of type $D_4$. A problem concerningthe number of conjugacy classes is also considered. The notion of nonseparablesubgroups plays a crucial role in all our constructions.

Complete reducibility, Kulshammer's question, conjugacy classes: a D_4 example