= 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 form
$$
\begin{pmatrix}A&*\\0&B\end{pmatrix},
$$
where $A$ 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 $L$.
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