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.
Show that there exists a basis of $E$ in which the matrices of elements of $\mathcal{A}$ belong to $\mathrm{T}_{n}^{+}(\mathbb{C})$.