Proposition 21

Let $G$ be a simply connected finite-dimensional solvable Lie group over $\mathbf{R}$ or $\mathbf{C}$ and $M$ an integral subgroup of $G$. Then $M$ is a Lie subgroup of $G$ and is simply connected.

We continue to use the notation $n, g, g_i, x_i, \phi$ of Proposition 20 but impose on the $x_i$ the following supplementary condition: let $i_p > i_{p-1} > \cdots > i_1$ be the integers $i$ such that $L(M) \cap g_i \neq L(M) \cap g_{i-1}$; then we take

\dagger Such a sequence exists by Chapter I, § 5, Proposition 2.

for $k = 1, 2, \ldots, p$. By induction on $n$, it is easily seen that $(x_{i_p}, x_{i_{p-1}}, \ldots, x_{i_1})$ is a basis of $L(M)$. Let $N$ be a simply connected Lie group such that there exists an isomorphism $h$ of $L(N)$ onto $L(M)$. Let

By Proposition 20, the mapping is an isomorphism of the manifold $K^p$ onto the manifold $N$. There exists a Lie group morphism $\tau$ of $N$ into $G$ such that $h = L(\tau)$ and $\tau(N) = M$ (§ 6, no. 2, Corollary 1 to Proposition 1). Hence $M$ is the set of elements of $G$ of the form Thus $M = \phi(T)$ where $T$ is a vector subspace of $K^n$.