  Let $G$ be a reductive algebraic group---possibly non-connected---over a
field $k$ and let $H$ be a subgroup of $G$. If $G= GL_n$ then there is a
degeneration process for obtaining from $H$ a completely reducible subgroup
$H'$ of $G$; one takes a limit of $H$ along a cocharacter of $G$ in an
appropriate sense. We generalise this idea to arbitrary reductive $G$ using the
notion of $G$-complete reducibility and results from geometric invariant theory
over non-algebraically closed fields due to the authors and Herpel. Our
construction produces a $G$-completely reducible subgroup $H'$ of $G$, unique
up to $G(k)$-conjugacy, which we call a $k$-semisimplification of $H$. This
gives a single unifying construction which extends various special cases in the
literature (in particular, it agrees with the usual notion for $G= GL_n$ and
with Serre's "$G$-analogue" of semisimplification for subgroups of $G(k)$). We
also show that under some extra hypotheses, one can pick $H'$ in a more
canonical way using the Tits Centre Conjecture for spherical buildings and/or
the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf
and Rousseau.
