Proposition 3

If a and b are ideals (resp. characteristic ideals) of g, [a, b] is an ideal (resp. a characteristic ideal) of g.

Let D be an inner derivation (resp. a derivation) of g. If $x \in a$ and $y \in b$, then

Hence the proposition.

If a is a submodule of g, the set of $x \in g$ such that $(\mathrm{ad}\ x).a \subset a$ is a subalgebra n of g called the normalizer of a in g. If further a is a subalgebra of g, then $a \subset n$ and a is an ideal of n.