Lie Automorphisms

Question

Take $X$ to be a Lie group. Define a Lie automorphism of $X$ to be a group isomorphism from $X$ to itself which is also a homeomorphism. Define $Aut(X)$ to be the group of Lie automorphisms of $X$ under composition. Then we can see that $Aut(X)$ is a subgroup of the homeomorphism group, $Homeo(X)$. Is there anything to be said of $Aut(X)$ as a subgroup?

Answer 1

First note that $Aut(X)$ is a subgroup of the diffeomorphism group of $X$.

In nice cases, for example if $\pi_0(X)$ is finite ($\pi_0(X)$ is the component group of $X$), then $Aut(X)$ is a Lie group itself. This is a theorem of Hochschild. (Paper: The Automorphism group of a Lie Group)

However in some certain cases one can say definitively more. For example, if $X=SU(2)$, then every automorphism is inner.

Do you have any concrete type of Lie groups in mind?