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 assume that $n \geqslant 2$ and that the result (that $\mathcal{A}$ is trigonalisable) is true for all natural integers $d \leqslant n-1$. We admit Burnside's Theorem: Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geqslant 2$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$. If the only vector subspaces of $E$ stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$, then $\mathcal{A} = \mathcal{L}(E)$. Show that there exists a vector subspace $V$ of $E$ distinct from $E$ and $\{0\}$ stable by all elements of $\mathcal{A}$.
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 assume that $n \geqslant 2$ and that the result (that $\mathcal{A}$ is trigonalisable) is true for all natural integers $d \leqslant n-1$.
We admit Burnside's Theorem: Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geqslant 2$. Let $\mathcal{A}$ be a subalgebra of $\mathcal{L}(E)$. If the only vector subspaces of $E$ stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$, then $\mathcal{A} = \mathcal{L}(E)$.
Show that there exists a vector subspace $V$ of $E$ distinct from $E$ and $\{0\}$ stable by all elements of $\mathcal{A}$.