Lie’s theorem

Let $𝔤$ be a finite dimensional complex solvable Lie algebra, and $V$ a repesentation of $𝔤$. Then there exists an element of $V$ which is a simultaneous eigenvector for all elements of $𝔤$.

Applying this result inductively, we find that there is a basis of $V$ with respect to which all elements of $𝔤$ are upper triangular.