Theorem 1

Let $u$ be an automorphism of a connected compact Lie group G.

a) The identity component of the group of fixed points of $u$ contains a regular element of G.

b) There exists a maximal torus K of G and a chamber of L(K) that are stable under $u$.

c) If G is simply-connected, the set of fixed points of $u$ is connected.

Assertion a) is the particular case $x=e$ of Prop. 3. We assume now that G is simply-connected and prove b) and c). Let $x$ be an element of G fixed by $u$, and let $g$ be a regular element of G, fixed by $u$ and commuting with $x$ (Prop. 3). The centralizer K of $g$ is a maximal torus of G (no. 2, Remark 2), stable under $u$, and containing $x$ and $g$. By Cor. 3 of Prop. 2 of no. 2, there exists a unique alcove A of L(K) such that $g\in$ exp A and $0\in A$; since $g$ is fixed by $u, L(u)$ leaves A, and hence also the chamber of L(K) containing A, stable. This proves b); further, the set of points of K fixed by $u$ is connected (Lemma 3) and contains $x$ and $e$, hence $c) ($General Topology, Chap. I, §11, no. 1, Prop. 2).

It remains to prove b) in the general case. Now, if $\widetilde{D}(G)$ is the universal covering of D(G), and if $f: \widetilde{D}(G)\rightarrow G$ is the canonical morphism, there exists an automorphism $\widetilde{u}$ of $\widetilde{D}(G)$ such that $f\circ \widetilde{u}=u\circ f$. If $\widetilde{K}$ is a maximal torus of $\widetilde{D}(G)$ and $\widetilde{C}$ a chamber of $L( \widetilde{K})$, stable under $\widetilde{u}$ (this exists by what has already been proved), there exists (§2, no. 3, Remark 2) a unique maximal torus K of G and a unique chamber C of L(K) such that $\widetilde{K} =f^{-1}(K)$ and $\widetilde{C} = L(f)^{-1}$(C), and we see immediately that K and C are stable under $u$, hence assertion b) in the general case.