Corollary

Let $V$ be a complex vector space of finite dimension $> 0$ and $G$ a connected solvable subgroup of $\mathbf{GL}(V)$. (i) There exists a non-zero element $v$ of $V$ such that $gv \in Cv$ for all $g \in G$. (ii) There exists a basis $B$ of $V$ such that, for all $g \in G$, the matrix of $g$ with respect to $B$ is lower triangular. Let $V_1$ be a minimal element among the non-zero vector subspaces of $V$ which are stable under $G$. By Proposition 22 (ii), $\dim V_1 = 1$. This proves (i). By induction on $\dim V$, it then follows that there exists an increasing sequence $(V_1, V_2, \ldots, V_n)$ of vector subspaces of $V$ which are stable under $G$ such that $\dim V_{i+1}/V_i = 1$ for $i < n$ and $V_n = V$; hence (ii).