Lie's theorem

Remarks

The proof is by induction on $\dim \mathfrak g$. Choose an ideal $\mathfrak h$ of codimension $1$, which exists since $[\mathfrak g, \mathfrak g] \ne \mathfrak g$, and a common eigenvector of $\mathfrak h$ with weight $\lambda$. The key lemma uses characteristic $0$ through a trace argument and shows that $\lambda$ vanishes on $[\mathfrak g, \mathfrak h]$. Then the weight space of $\lambda$ is stable under all of $\mathfrak g$. An eigenvector in it of an element outside $\mathfrak h$ does the job. The triangular form follows by applying the first part to $V / kv$ and inducting on $\dim V$.

Both hypotheses on $k$ are needed. Over $\R$, the abelian Lie algebra spanned by the rotation generator $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ acting on $\R^2$ has no real eigenvector. In characteristic $p > 0$, let $V = k^p$ with basis $e_0, \dots, e_{p-1}$, let $x e_i = i e_i$ and $y e_i = e_{i+1}$ with indices mod $p$. Then $[x, y] = y$, so $x$ and $y$ span a solvable Lie algebra. The eigenvectors of $x$ are the nonzero multiples of the $e_i$, and none of them is an eigenvector of $y$.

Going deeper

For a student

The theorem shows that the irreducible finite-dimensional representations of a solvable Lie algebra over an algebraically closed field of characteristic $0$ are one-dimensional. Applied to the adjoint representation, it gives a chain of ideals of every dimension and shows that $[\mathfrak g, \mathfrak g]$ is nilpotent, using Engel's theorem. It is a key step toward Cartan's criterion, the structure theory of semisimple Lie algebras and the conjugacy of Borel subalgebras. The analogue for connected solvable linear algebraic groups is the Lie–Kolchin theorem.