The center of a nilpotent Lie algebra intersects each ideal
Solution 1:
Let $I$ be a nontrivial ideal of $L$. Then $L$ acts on $I$ by the adjoint action, because $I$ is an ideal. Then by Engel's theorem (or lemma to it) there exists a $v\neq 0$ in $I$ with $0=L.v=[L,v]$, because $L$ is nilpotent. But this just means that $$ v\in I\cap Z(L), $$ so that the intersection is nontrivial. In particular, the center of a nilpotent Lie algebra itself is nontrivial.