Proposition 28

Let G be a simply connected real or complex Lie group. The following conditions are equivalent: (i) G is semi-simple; (ii) G is isomorphic to the product of a finite number of almost simple groups. If G is a finite product of almost simple Lie groups, $L(G)$ is a finite product of simple Lie algebras and is hence semi-simple. If G is semi-simple, $L(G)$ is isomorphic to a product of simple Lie algebras $L_1, \ldots, L_n$. Let $G_i$ be a simply connected Lie group with Lie algebra $L_i$, which is therefore almost simple. Then G and $G_1 \times \cdots \times G_n$ are simply connected and have isomorphic Lie algebras and are therefore isomorphic.