  Let $G$ be a connected reductive algebraic group defined over an
algebraically closed field %$k$ of characteristic $p > 0$. Our first aim in
this note is to give concise and uniform proofs for two fundamental and deep
results in the context of Serre's notion of $G$-complete reducibility, at the
cost of less favourable bounds. Here are some special cases of these results:
Suppose that the index $(H:H^\circ)$ is prime to $p$ and that $p > 2\dim V-2$
for some faithful $G$-module $V$. Then the following hold: (i) $V$ is a
semisimple $H$-module if and only if $H$ is $G$-completely reducible; (ii)
$H^\circ$ is reductive if and only if $H$ is $G$-completely reducible.
  We also discuss two new related results: (i) if $p \ge \dim V$ for some
$G$-module $V$ and $H$ is a $G$-completely reducible subgroup of $G$, then $V$
is a semisimple $H$-module -- this generalizes Jantzen's semisimplicity theorem
(which is the case $H = G$); (ii) if $H$ acts semisimply on $V \otimes V^*$ for
some faithful $G$-module $V$, then $H$ is $G$-completely reducible.
