The center of nilpotent Lie algebra and the last abelian term of the derived series

Question

Let $L$ be a finite-dimensional nilpotent lie algebra. Consider the last non-trivial term of the derived series $A:=[L^n,L^n]$. Is it true that $A$ is equal to the center $Z$ of $L$?

Answer 1

No, consider any nilpotent algebra $L$ which is not commutative. Let $M$ be a commutative algebra. Suppose that $L^n\neq 0$ and $L^{n+1}=0$. The center of $L\oplus M$ is $Z(L) \oplus M$, and$(L\oplus M)^n=L^n$, $(L\oplus M)^{n+1}=0$.