Theorem 1

Let $G$ be a finite-dimensional real or complex Lie group, $G_0$ its identity component, $R$ its radical and $r$ the radical of $L(G)$; suppose that $G/G_0$ is finite. Let $\rho$ be a finite-dimensional analytic linear representation of $G$. The following conditions are equivalent: (i) $\rho$ is semi-simple; (ii) $\rho|G_0$ is semi-simple; (iii) $\rho|R$ is semi-simple; (iv) $L(\rho)$ is semi-simple; (v) $L(\rho)|r$ is semi-simple. (i) $\Leftrightarrow$ (ii) by Lemma 2 and Integration, Chapter VII, § 3, Proposition 1. (ii) $\Leftrightarrow$ (iv) and (iii) $\Leftrightarrow$ (v) by § 6, no. 5, Corollary 2 to Proposition 13. (iv) $\Leftrightarrow$ (v) by Chapter I, § 6, Theorem 4.