I am wondering whether the following claim is true:
Let $G$ be a Lie group, $\mathfrak{g}$ its Lie algebra and $V$ some vector subspace of $\mathfrak{g}$.
Claim: $V$ is an ideal of $\mathfrak{g}$ if and only if $V$ is invariant under $\mathrm{Ad} \colon G \to \mathrm{GL}(\mathfrak{g})$, i.e. for all $g \in G$
$$ \mathrm{Ad}_g(V) \subseteq V $$
It is clear to me, that $V$ has to be invariant under $\mathrm{ad} \colon \mathfrak{g} \to \mathrm{End}(\mathfrak{g})$, since $\mathrm{ad}_X(Y)= [X,Y]$, but I struggle to proof this for $\mathrm{Ad}$. It may be that the claim only holds for semisimple Lie algebras, but I am not sure about that.
Any help or references are greatly appreciated!