Proposition 26

Let G be a finite-dimensional connected real or complex Lie group. The following conditions are equivalent: (i) $L(G)$ is semi-simple; (ii) the radical of G is $\{e\}$; (iii) every normal commutative integral subgroup of G is equal to $\{e\}$. Condition (ii) means that the radical of $L(G)$ is $\{0\}$ and hence (i) $\Leftrightarrow$ (ii) (Chapter I, § 6, Theorem 1). The equivalence of (i) and (iii) follows from § 6, no. 6, Proposition 14.