Proposition 20

Let $G$ be a simply connected solvable Lie group of finite-dimension $n$ over $\mathbf{R}$ or $\mathbf{C}$ and $g = L(G)$. Let $(g_n, g_{n-1}, \ldots, g_0)$ be a sequence of subalgebras of $g$ of dimensions $n, n-1, \ldots, 0$, such that $g_{i-1}$ is an ideal of $g_i$ for $i = n, n-1, \ldots, 1.\dagger$ Let $G_i$ be the integral subgroup of $G$ corresponding to $g_i$. Let $x_i$ be a vector of $g_i$ not belonging to $g_{i-1}$. Let $\phi_i$ be the mapping

of $K^i$ into $G$. Then $\phi_n$ is an isomorphism of analytic manifolds and $\phi_i(K^i) = G_i$ for all $i$.

For $n = 0$ the proposition is obvious. We argue by induction on $n$. Let $H$ be the integral subgroup of $G$ such that $L(H) = Kx_n$. By § 6, no. 6, Corollary 1 to Proposition 14, $H$ and $G_{n-1}$ are simply connected Lie subgroups of $G$ and, as a Lie group, $G$ is the semi-direct product of $H$ by $G_{n-1}$. Hence $\lambda \mapsto \exp(\lambda x_n)$ is an isomorphism of $K$ onto $H$ and by the induction hypothesis the mapping

is an isomorphism of the analytic manifold $K^{n-1}$ onto the analytic manifold $G_{n-1}$ which maps $K^i \times \{0\}$ to $G_i$ for $i = 1, 2, \ldots, n-1$. Hence the proposition.