Exercise 9, § 2

Let G be a connected finite dimensional real Lie group. Let $\mathfrak{g}$ be its Lie algebra, and let $\{x_1, . . . , x_n\}$ be a generating family of $\mathfrak{g}$. For $m\geq 0$, denote by $\Gamma_m$ the subgroup of G generated by the exp(2$^{-m}x_i$)$, 1\leq i\leq n$. We have $\Gamma_0\subset \Gamma_1\subset \cdots$.

a) Show that the union of the $\Gamma_m$ is dense in G.

b) Let $H_m$ be the identity component of the closure $\overline{\Gamma}_m$ of $\Gamma_m$. We have $H_0\subset H_1\subset \cdots$, and the family $(H_m)$ is stationary; let H be the common value of the $H_m$ for $m$ sufficiently large. Show that H is normal in G (observe that H is normalized by all the $\Gamma_m$), and that the image of $\Gamma_m$ in $G/H$ is a discrete subgroup of $G/H$. The union of these subgroups being dense in $G/H$, deduce (cf. Chap. III, §6, Exerc. $23d$)$)$ that $G/H$ is nilpotent.

c) Assume that $\mathfrak{g}=\mathscr{D}\mathfrak{g}$. Show that G = H, in other words, that $\Gamma_m$ is dense in G if $m$ is sufficiently large.