non nilpotent Lie algebras

Question

Let $L$ be a nonperfect Lie algebra (i.e. $L \ne [L, L]$) which is not nilpotent. Is there a ideal of $L$ such as $M$ such that intersection drived subalgebra of $L$ and $Z(M)$ is nonzero?

Answer 1

Yes, there is such an ideal. Take $L$ to be the $2$-dimensional nonabelian Lie algebra with basis $e_1,e_2$ and Lie bracket $[e_1,e_2]=e_1$ and the ideal $M=\langle e_1\rangle$. Then $L$ is not nilpotent, $L\neq [L,L]$ and $M$ is an ideal with $Z(M)=[L,L]\neq 0$.

In general, however, this need not be true. Take $L=\mathfrak{gl}_2(\Bbb{C})=\mathfrak{sl}_2(\Bbb{C})\oplus \Bbb{C}$. Then $L$ is not perfect and not nilpotent, but $Z(M)$ is either zero or $\Bbb{C}$ for all ideals $M$, and the intersection with $[L,L]$ is zero.