Lie algebra adjoint representation

Question

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Then there are representations $$Ad : G \rightarrow GL(\mathfrak{g}), \; \; ad : \mathfrak{g} \rightarrow GL(\mathfrak{g}).$$ Subrepresentations of $ad$ are ideals of $\mathfrak{g}$ and are interesting. Is there a name for subrepresentations of $Ad$? (in fact, are these the same ideals? how to see this?)

Answer 1

Yes, follows from the section "An important lemma" here.

Answer 2

I think it is a bit more instructive to look at the adjoint representation as follows. Let $G$ be a compact Lie group and $\mathfrak{g}$ it's Lie algebra. Identify $\mathfrak{g}$ with left invariant vector fields. Now $G$ acts on $\mathfrak{g}$ on both the right and left. More precisely, we define for all $h\in \mathfrak{g}$, $g\cdot h=L_{g *}h$ and $h\cdot g=R_{g *}h$. Now the we can describe the adjoint representation on $\mathfrak{g}$ as the action of $G$ on $\mathfrak{g}$ given by taking $h\in \mathfrak{g}$ to $g\cdot h \cdot g^{-1}$. Then the subrepresentations of the adjoint representation are the subspaces $V$ of $\mathfrak{g}$ such that $gVg^{-1}=V$for all $g\in G$ (normal with respect to this left and right action).

As for the ideals of $\mathfrak{g}$ being subrepresentations of $ad$, you can use the fact that $ad_{x}(y)=[x,y]$. So a subrepresentation of $ad$ is a subspace that is closed under the ring multiplication $[x,y]$ in $\mathfrak{g}$, hence an ideal.