Show that if $L$ is a nilpotent $n$-dimensional Lie algebra, then there is an ideal of dimension $n-1$

Question

This is my first question in the forum so I’m sorry for any mistake. $\ddot\smile$

I had a test in Lie algebras course and there was a question that asked for existence of ideal.

Let $L$ be a lie algebra of dimension $n$ such that $L$ is nilpotent. Show that there exist an ideal $J$ of $L$ such that $\dim J = n-1$.

I’d tried to work with the definitions but I couldn’t find a way to prove the dimension of the ideal.

Answer 1

Since $L$ is nilpotent, $[L,L]\varsubsetneq L$. So, take any subspace $J$ of $L$ whose dimension is $n-1$ such that $[L,L]\subset J$ and then $J$ will be an ideal of $L$ whose dimension is $n-1$.