Subalgebra condition in Engel's theorem

Question

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 $L\cdot v = 0$?

Answer 1

You're completely right, maybe it's a matter of taste.

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.

It's similar for Lie's Theorem: Before proving the representation theoretic version, one shows that solvable subalgebras of matrices admit common eigenvectors (over an algebraically closed field) and deduces the other formulations afterwards.