{\rm rank} \ \! H$, then a maximal torus in $K$ is group-conjugate with a subgroup of $H$.

If one of $\sigma$ or $\tau$ is an inner automorphism, the claim in the lemma is clear by [@He Thm. 5.6, p. 424]. From now on we will assume that both $\sigma$ and $\tau$ are outer automorphisms of $G$. Let $T_K\subset K$ be an arbitrary maximal torus. There is a unique maximal torus $T$ in $G$ such that $T_K\subset T$; furthermore, $T$ is $\tau$-invariant and there is a Weyl chamber $C$ in ${\mathfrak t}:={\rm Lie}(T)$ which is $\tau$-invariant as well (see [@Lo Prop. 3.2, p. 125]). Let $c: G \to G$ be an inner automorphism such that the torus $S:=c(T)$ contains $T_H$, the latter being a maximal torus in $H$. As before, there exists a Weyl chamber $C'\subset c({\mathfrak t})$ which is invariant under $\sigma$. On the other hand, $c\tau c^{-1}$ is an involutive automorphism of $G$ which leaves $S$ invariant; it even leaves the chamber $c(C)$ (inside the Lie algebra of $S$) invariant. But the chambers $c(C)$ and $C'$ are conjugate under the Weyl group of $(G,S)$; that is, there exists an inner automorphism, say $c'$, such that $$c'(S) =S \ 