Corollary 2

Let $\Gamma$ be a compact subgroup of Aut(G) with the following property:

(*) There exist elements $u_1, . . . , u_n$ of $\Gamma$ such that, for all $i$, the closure $\Gamma_i$ of the subgroup of $\Gamma$ generated by $u_1, . . . , u_i$ is a normal subgroup of $\Gamma$, and $\Gamma_n=\Gamma$.

Then, there exists a maximal torus of G stable under the operation of $\Gamma$.

We argue by induction on the dimension of G. Clearly, we can assume that $u_1\not=$ Id; then the subgroup H of fixed points of $u_1$ is distinct from G, and is stable under the operation of $\Gamma$. Moreover, since $\Gamma$ is compact, the image of $\Gamma$ in Aut(H$_0$) is a quotient of $\Gamma$, hence it also satisfies condition (*). By the induction hypothesis, there exists a maximal torus S of H stable under $\Gamma$. The centralizer K of S in G is connected (§2, no. 2, Cor. 5) and stable under $\Gamma$; this is a maximal torus of G, since $H_0$ contains a regular element of G (Th. $1a$)$)$ which is conjugate to an element of S (loc. cit., Cor. 4).