\textbf{Extension to $V$ compatible with $u$}\\
Let $V$ be a finite-dimensional vector space equipped with a nilpotent endomorphism $u$. We assume that there exists a vector subspace $W$ of $V$ stable by $u$ and a linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. Prove that $\varphi$ admits an extension $\psi$ to $V$ compatible with $u$.