Corollary 2 (Lie’s Theorem)

Let $g$ be a solvable Lie algebra; suppose that $K$ is algebraically closed. Let $M$ be a $g$-module of finite dimension over $K$ and let $(M_i)_{0 \leq i \leq r}$ be a Jordan-Hölder series of $M$. Then $M_{i-1}/M_i$ is of dimension 1 over $K$ for $1 \leq i \leq r$ and, for all $x \in g$, $x_{M_{i-1}/M_i} = \lambda_i(x) \cdot 1$, where $\lambda_i$ is a linear form on $g$ which is zero on $\mathscr{D}g$. In particular, every simple $g$-module of finite dimension over $K$ is in fact of dimension 1.

Let $\rho_i$ be the representation of $g$ on $M_{i-1}/M_i$. The associative algebra $L_i$ generated by 1 and $\rho_i(g)$ is a field, a finite extension of $K$ and therefore equal to $K$; and $M_{i-1}/M_i$ is a simple $L_i$-module, whence $\dim M_{i-1}/M_i = 1$. The rest of the corollary is obvious.