Proposition 1

Let G and $G'$ be two connected compact Lie groups.

a) Let $f: G\rightarrow G'$ be a surjective morphism of Lie groups. The maximal tori of $G'$ are the images under $f$ of the maximal tori of G. If the kernel of $f$ is central in G (for example discrete), the maximal tori of G are the inverse images under $f$ of the maximal tori of $G'$.

b) Let H be a connected closed subgroup of G. Every maximal torus of H is the intersection with H of a maximal torus of G.

c) Let H be a connected closed normal subgroup of G. The maximal tori of H are the intersections with H of the maximal tori of G.

a) Let T be a maximal torus of G; then L(T) is a Cartan subalgebra of L(G) (no. 2, Th. $2a$)$)$, so $L(f(T))$ is a Cartan subalgebra of $L(G')$ (Chap. VII, §2, no. 1, Cor. 2 of Prop. 4); it follows that $f(T)$ is a maximal torus of $G'$ (no. 2, Th. $2a$)$)$. If Ker $f$ is central in G, it is contained in T (Cor. 2 of Th. 2), so $T =f^{-1}(f(T))$.

Conversely, let $T'$ be a maximal torus of $G'$; we show that there exists a maximal torus T of G such that $f(T) = T'$. Let $T_1$ be a maximal torus of G; then $f(T_1)$ is a maximal torus of $G'$ and there exists $g'\in G'$ such that $T'=g'f(T_1)g^{'-1}$ (Th. $2b$)$)$; if $g\in G$ is such that $f(g) =g'$, we have $T'=f(T)$ with $T =gT_1g^{-1}$.

b) Let S be a maximal torus of H; this is a torus of G so there exists a maximal torus T of G containing S. Then $T\cap H$ is a commutative subgroup of H containing S, hence is equal to S (no. 2, Remark 2).

c) By §1, no. 3, Prop. $2c), L(G)$ is the direct product of L(H) with an ideal; thus, the Cartan subalgebras of L(H) are the intersections with L(H) of the Cartan subalgebras of L(G). Thus, for any maximal torus T of $G, T\cap H$ contains a maximal torus S of H and $S = T\cap H$ (no. 2, Remark 2).