Proposition 4

Let G be a connected Lie group whose Lie algebra is compact. There exist a torus T, a simply-connected compact semi-simple Lie

$^4$ Note that a real topological vector space cannot be a compact topological space

unless it is reduced to 0. group S, a vector group V and a surjective morphism $f: V\times T\times S\rightarrow G$ with finite kernel. If G is compact, the group V is reduced to the identity element.

Let C (resp. S) be a simply-connected Lie group whose Lie algebra is isomorphic to the centre (resp. the derived algebra) of L(G). Then C is a vector group, S is a compact group with finite centre (Th. 1) and G can be identified with the quotient of $C\times S$ by a discrete subgroup D, which is central (Integration, Chap. VII, §3, no. 2, Lemma 4). Since the image of the projection of D onto S is central, hence finite, $D\cap C$ is of finite index in D. Let $C'$ be the vector subspace of C generated by $D\cap C$, and V a complementary subspace. Then the group $T = C'/(D\cap C)$ is a torus, and G is isomorphic to the quotient of the product group $V\times T\times S$ by a finite group.

If G is compact, so is $V\times T\times S ($General Topology, Chap. III, §4, no. 1, Cor. 2 of Prop. 2), hence so is V, which implies that $V =\{e\}$.