= Simultaneous triangularization of a Lie algebra representation
{title2=$0=V_0\subset V_1\subset\cdots\subset V_n=V$}
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.
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