Corollary

Let $\mathfrak{s}$ be a compact Lie algebra, and let $\varphi$ be an automorphism of $\mathfrak{s}$. There exists a regular element of $\mathfrak{s}$ fixed by $\varphi$.

Replacing $\mathfrak{s}$ by $\mathscr{D}\mathfrak{s}$, we can assume that $\mathfrak{s}$ is semi-simple. Let S be a compact simply-connected Lie group with Lie algebra $\mathfrak{s}$, and let $u$ be the automorphism of S such that $L(u) =\varphi$. Prop. 3 implies the existence of an element $a$ of $\mathfrak{s}$, fixed by $\varphi$, such that exp $a$ is regular in S; in particular, $a$ is regular in $\mathfrak{s}$ (no. 2, remark 4).