Corollary 1

Let G be a connected compact Lie group. Then $C(G)_0$ is a torus, D(G) is a connected compact semi-simple Lie group and the morphism $(x, y) \rightarrow xy$ from $C(G)_0\times D(G)$ to G is a finite covering.

With the notation in Prop. 4, we have $V =\{e\}$ and the subgroups $f(T)$ and $f(S)$ of G are compact, hence closed. Thus it suffices to show that $f(T) =$ $C(G)_0,f(S) = D(G)$. Now, $L(G) = L(f(T))\times L(f(S))$; since S is semi-simple and T is commutative, this implies that $L(f(T)) =\mathscr{C}(L(G)) = L(C(G)_0)$ (Chap. III, §9, no. 3, Prop. 8) and $L(f(S)) =\mathscr{D}L(G) = L(D(G))$ (Chap. III, §9, no. 2, Cor. of Prop. 4), hence the stated assertion.