 Solution

A vector subspace $I\subseteq \mathfrak{g}$ is an ideal of a Lie algebra when $[\mathfrak{g},I]\subseteq I$. A nilpotent Lie algebra is one whose lower central serieseventually vanishes.

By the Engel theorem, the operators ${\mathrm{ad}}_x$ for a nilpotent complex Lie algebra can be represented simultaneously by strictly upper triangular matrices. Their products are strictly upper triangular and have zero trace. Hence