Theorem 1 (Engel)

Let $V$ be a vector space over $\mathbf{K}$ and $g$ a finite-dimensional subalgebra of $\mathfrak{gl}(V)$ whose elements are nilpotent endomorphisms of $V$. If $V \neq \{0\}$, there exists $u \neq 0$ in $V$ such that $x.u = 0$ for all $x \in g$.

The proof proceeds by induction on the dimension $n$ of $g$. The theorem is obvious if $n = 0$. Suppose that it is true for algebras of dimension $< n$.

Let $h$ be a Lie subalgebra of $g$ of dimension $m < n$. If $x \in h$, and $\mathrm{ad}_g x$ maps $h$ into itself and defines on passing to the quotient an endomorphism $\sigma(x)$ of the space $g/h$. By Lemma 1 $\mathrm{ad}_g x$ is nilpotent and hence $\sigma(x)$ is nilpotent. By the induction hypothesis there exists a non-zero element of $g/h$ which is annihilated by all the $\sigma(x),\ x \in h$. It follows that $h$ is an ideal in a certain $(m+1)$-dimensional subalgebra of $g$.

We conclude (by iteration starting with $h = \{0\}$) that $g$ has an ideal $h$ of dimension $n-1$. Let $a \in g,\ a \notin h$. We again use the induction hypothesis: the $u \in V$ such that $x.u = 0$ for all $x \in h$ form a non-zero vector subspace U of V. This subspace is stable under $a$ (§ 3, no. 5, Proposition 5). Since $a$ is a nilpotent endomorphism of V, there exists a non-zero element of U which is annihilated by $a$ and hence by every element of $g$.