Proposition 5

Let $V$ be a vector space of finite dimension $n$ over $K$ and $g$ a Lie sub-algebra of $gl(V)$ whose elements are nilpotent endomorphisms of $V$. Then there exists a decreasing sequence of vector subspaces $V_0, V_1, \ldots, V_n$ of $V$, of dimensions $n, n-1, \ldots, 0$, such that $x(V_i) \subset V_{i+1}$ for all $x \in g$ and $i = 0, 1, \ldots, n-1$.