lie algebras - reductive Lie subalgebra

Sorry, this started as a comment, but got too long.

If $G$ is semisimple, then every derivation of $G$ is inner, so
that the normalizer $N_L(G)=C_L(G)+G$ where $C_L(G)$
is the centralizer. In this situtaion Michele's condition holds
if and only if the centralizer is trivial.

However the case where $G$ is reductive isn't so easy.
A reductive $G$ is the direct sum of an Abelian and a semisimple
Lie algebra. The Abelian part can have (if at least two-dimensional)
non-trivial derivations. These will lift to non-inner derivations
of $G$. If we let $L$ be the semidirect product of $G$ with
a one-dimensional Lie algebra using this derivation, then
$L$ normalizes $G$, $Lne G$ but $C_L(G)$ is trivial.