Let $E$ be a $\mathbb{C}$-vector space of finite dimension $n \geqslant 1$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$ consisting of nilpotent endomorphisms. We propose to prove by strong induction on $n \in \mathbb{N}^{*}$ that if all elements of $\mathcal{A}$ are nilpotent, then $\mathcal{A}$ is trigonalisable.
Show that the result is true if $n = 1$.