Given a Lie group $G$ with finitely many components and a compact Lie group A which acts on $G$ by automorphisms, we prove that there always exists an A-invariant maximal compact subgroup K of G, and that for every such K, the natural map $H^1(A,K)to H^1(A,G)$ is bijective. This generalizes a classical result of Serre [6] and a recent result in [1].
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