So I am just making my way into the theory of Lie Algebras. The question at hand comes from page 14 of Humphreys, Introduction to Lie Algebra and Representation Theory.
Given a finite dimensional nilpotent Lie algebra $L$, and a proper sub-algebra $K$ of $L$, prove that the normalizer of $K$ properly contains $K$.
I keep trying to use Engel's theorem, but I have only been able to show there is a vector such that $[v\ K]$ is in the center, which much be in $K$. Though the condition for $v$ was only that it is in $L$.
A proof or anything else would be helpful. Thank you.