Proposition 31

Let G be a simply connected finite-dimensional real or complex Lie group and R its radical. There exists a semi-simple simply connected Lie subgroup S of G such that G, as a Lie group, is the semi-direct product of S by R. If S' is a semi-simple integral subgroup of G, there exists x in the nilpotent radical of $L(G)$ such that

This follows from § 6, no. 6, Corollary 1 to Proposition 14 and Chapter I, § 6, Theorem 5 and Corollary 1.