# Subalgebra condition in Engel's theorem

An equivalent version of Engel's theorem says that

Let $L$ be a subalgebra of $\mathfrak{gl}(V)$, $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V\ne 0$, then there exists a nonzero $v\in V$ for which $L v = 0$.

Why do we need this condition that $\rho:L\hookrightarrow \mathfrak{gl}(V)$? If this representation is not faithful can't we quotient out the kernel $\overline{\rho}:L/\ker\rho\to \mathfrak{gl}(V)$ and conclude that $(L/\ker\rho)\cdot v = 0$ and therefore $L\cdot v = 0$?

One benefit of the formulation you quote is that it is very elementary: it doesn't speak about abstract representations of Lie algebras, but only about spaces of nilpotent matrices closed under commutator - pure linear algebra, if you want. However, as far as I remember, for the usual inductive proof one needs to change both ${\mathfrak g}$ and $V$ and consider representations which are not necessarily faithful, so your version of the theorem fits that induction better than the quoted one, and I'd therefore would prefer it, too.