Simultaneous triangularization of a Lie algebra representation ($0=V_0\subset V_1\subset \cdots \subset V_n=V$)

A Lie algebra representation is simultaneously upper triangular precisely when it preserves a complete flag, with $\dim V_j=j$. A basis adapted to the invariant flag makes every representing matrix upper triangular. For a complex finite-dimensional representation of a Solvable Lie algebra, the Lie theorem provides a common eigenvector; iteration on the quotient representations constructs the flag.