How is the automorphism group of a Lie group given a differential structure?
Solution 1:
The fact that the automorphism group $Aut({\mathfrak g})$ of a Lie algebra ${\mathfrak g}$ is a Lie group is not immediate. However, it follows from Cartan's Theorem that every closed subgroup of a Lie group is a Lie subgroup. In your case, $Aut({\mathfrak g})$ is a subgroup of $GL({\mathfrak g})$ given by the condition that you have mentioned. This condition clearly defines a closed subset of $GL({\mathfrak g})$ from which it follows that $Aut({\mathfrak g})$ is a Lie subgroup of $GL({\mathfrak g})$.