Exercise 20, § 9

Let G be a finite-dimensional solvable connected real Lie group. We introduce the notation S, S', F, $\sigma$ of § 6, no. 10, proof of Proposition 20.

(a) Using Proposition 21, show that $\sigma$ is an isomorphism of S onto a Lie subgroup of the underlying real Lie group of S'.

(b) Deduce that the universal complexification $\tilde{G}$ of G is identified with $S'/\sigma(F)$ and that the canonical mapping of G into $\tilde{G}$ is an isomorphism of G onto a Lie subgroup of the underlying real Lie group of $\tilde{G}$.