  Let $G$ be a connected reductive linear algebraic group over a field $k$.
Using ideas from geometric invariant theory, we study the notion of
$G$-complete reducibility over $k$ for a Lie subalgebra $\mathfrak h$ of the
Lie algebra $\mathfrak g = Lie(G)$ of $G$ and prove some results when
$\mathfrak h$ is solvable or $char(k)= 0$. We introduce the concept of a
$k$-semisimplification $\mathfrak h'$ of $\mathfrak h$; $\mathfrak h'$ is a Lie
subalgebra of $\mathfrak g$ associated to $\mathfrak h$ which is $G$-completely
reducible over $k$. This is the Lie algebra counterpart of the analogous notion
for subgroups studied earlier by the first, third and fourth authors. As in the
subgroup case, we show that $\mathfrak h'$ is unique up to $Ad(G(k))$-conjugacy
in $\mathfrak g$. Moreover, we prove that the two concepts are compatible: for
$H$ a closed subgroup of $G$ and $H'$ a $k$-semisimplification of $H$, the Lie
algebra $Lie(H')$ is a $k$-semisimplification of $Lie(H)$.
