Proposition 29

Let G be a semi-simple connected real or complex Lie group. (i) $G = (G, G)$. (ii) The centre Z of G is discrete. (iii) The centre of $G/Z$ is $\{e\}$. Assertion (i) follows from the Corollary to Proposition 4, no. 2 and Chapter I, § 6, Theorem 1. Assertion (ii) follows from § 6, no. 4, Corollary 4 to Proposition 10 and Chapter I, § 6, no. 1, Remark 2. Assertion (iii) follows from (ii) and Lemma 1.