We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$.
Assume that $\operatorname{Im} u = \operatorname{Ker} u$. Show that there exist vectors $e_1, e_2, \ldots, e_r$ in $E$ such that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right)\right)$ is a basis of $E$.