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Clarified question

Does Aut(G) → Out(G) always split for a compact, connected Lie group G?

The outer automorphism group of a topological group $G$ is constructed by the short exact sequence $$ 1\longrightarrow \operatorname{Inn}(G) \longrightarrow \operatorname{Aut}(G) \longrightarrow \operatorname{Out}(G) \longrightarrow 1. $$ This sequence does not always split, see Non-split Aut(G) $\to$ Out(G)?, for example for the disrete group $G = A_6$.

I am interested in the case where $G$ is a compact, connected Lie group. Does the sequence always split in this case? (If $G$ has a simple Lie algebra $\mathfrak{g}$ then I believe the answer is yes.)