Every Lie algebra contains a maximal proper Lie subalgebra

Question

I am working though the proof of Proposition 6.2 in Erdmann's "Introduction to Lie Algebras".

I can't verify that every Lie subalgebra of $L \subseteq \mathfrak{gl}(V)$ contains a maximal (proper) Lie subalgebra.

How would I prove this fact?

Answer 1

If the dimension of $V$ is finite, consider $S(L)$ the of Lie subalgebras distinct of $L$, it is not empty since it contains $0$, an element of maximum dimension is a maximum proper Lie subalgebra.