Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2007 / A / 1 / c / Solution

Start with the given basis of $U$, and follow it by arbitrary lifts of the given basis of $V/U$. Since $U$ is an invariant linear subspace, every acting endomorphism has block formwhere $A$ and $B$ are its matrices on $U$ and $V/U$. Both are strictly upper triangular in the chosen bases. Therefore the whole matrix has zero entries on and below its diagonal. The same lifted basis works simultaneously for all elements of $L$.