If $\frak{g}$ is a Lie algebra of finite dimension, why is $\operatorname{Aut}(\frak{g})$ a closed subgroup of $GL(\frak{g})$?

Question

If $\frak{g}$ is a Lie algebra of finite dimension, why is $\operatorname{Aut}(\frak{g})$ a closed subgroup of $GL(\frak{g})$?

Could someone help me answer this question?

Answer 1

Because$$\operatorname{Aut}(\mathfrak{g})=\bigcap_{X,Y\in\mathfrak g}\left\{g\in GL(\mathfrak{g})\,\middle|\,g[X,Y]=[gX,gY]\right\}.$$This expresses $\operatorname{Aut}(\mathfrak{g})$ as an intersection of closed sets.