How could I show that $\mathrm{Aut}(\mathcal{A})$ is the isotropy group of some point in some representation of $GL(\mathcal{A})$?

Question

Let $\mathcal{A}$ be a Lie algebra, and $$ \operatorname{Aut}(\mathcal{A})=\{a \in \operatorname{End}(\mathcal{A}) \mid[a(x), a(y)]=a([x, y]), \forall x, y \in \mathcal{A}\}, $$ where $\operatorname{End}(\mathcal{A})$ denotes the set of linear maps $\mathcal{A}\to\mathcal{A}$. I have to show that $\operatorname{Aut}(\mathcal{A})$ is a closed Lie subgroup of $GL(\mathcal{A})$, and I've been provided the hint that I should first show that $\mathrm{Aut}(\mathcal{A})$ is the isotropy group (a.k.a. the little group or the stabilizer) of some point in some representation of $GL(\mathcal{A})$. Could someone give me a sketch or clue of how to prove the latter?