This natural map is a representation. If K is a K invariant vector subspace of L under ad, then we are done by the previous lemma. Then L is nilpotent. Then by lemma, L is nilpotent.
L is ad-nilpotent.
Suppose V is a vector space. Let L be a nilpotent Lie algebra. Commuting matrices correspond to an abelian Lie algebra. We must prove this for solvable Lie algebras, which will imply the other case. Then all endomorphisms in L are multiples of one endomorphism, thus they all have the same eigenvectors.