nLab automorphism Lie group

The automorphism group $Aut(G)$ of a Lie group $G$ itself canonically inherits the structure of a Lie group. Its Lie algebra is the derivation Lie algebra of the Lie algebra $\mathfrak{g}$ of $G$: $Lie(Aut(G)) \simeq \mathfrak{der}(\mathfrak{g})$.