The set of $N_L(G)/G$ has trivial center, where l$such that$N_L(G)$[l,g]\in G$ for all $G$ is called the normalizerin $L$ of $G$. It's not particularly common for the normalizer to be the same as G, though it does happen sometimes. There are a few theorems I know about specific Lie algebras being self-normalizing, such as Cartan and Borel subalgebras in reductive Lie algebras, but it's not very common and I don't know any general condition which guarantees it.
This is true if and only if $L/G$ N_L(G)/G$has trivial center., where$N_L(G)$is the normalizer in$L$of$G$. 1 This is true if and only if$L/G\$ has trivial center.