Proposition 3

Let $u$ be an automorphism of G, and let $x$ be a point of G fixed by $u$.

a) There exists an element $a$ of $\mathfrak{g}$, fixed by $L(u)$ and by Ad $x$, such that $x$ exp $a$ is regular.

b) There exists a regular element $g$ of G fixed by $u$ and commuting with $x$.

Let H be the group of fixed points of $u, S$ a maximal torus of $Z(x)\cap H$, and K the identity component of $Z(S)\cap Z(x)$. This is a connected closed subgroup of G; further, by Cor. 5 of Th. 2 of §2, no. 2, there exist maximal tori of G containing S and $x$, so K is of maximal rank and contains S and $x$. On the other hand, K is stable under $u$ since S and $x$ are; denote by V the set of fixed points of $u$ in K. Then

so $V_0$ is contained in the centralizer of S in $(Z(x)\cap H)_0$; but the latter reduces to S (loc. cit., Cor. 6), hence finally $V_0= S$. Lemma 1 now implies that K is commutative, hence is a maximal torus of G (since it is connected and of maximal rank). It contains S and $x$, and is equal to the identity component of $Z(S)\cap Z(x)$; assertion a) now follows from Lemma 2. We deduce b) by taking $g=x$ exp $a$.