Transferred from MSE where it now received a complete answer. Maybe the following is easy, but I am not an expert in finite-dimensional Lie algebras and was stuck on the following problem.

Can you prove or disprove the following statement ?

S)Let $V$ be an indecomposable (finite-dimensional) module over a nilpotent $k$-Lie algebra ($k$ is algebraically closed), then there is a unique character $c: {\frak{g}}\to k$ ($c$ is linear and $[{\frak{g}},{\frak{g}}]\subset ker(c)$) such that, for all $g\in \frak{g}$, the operator $$ \pi(g)-c(g)Id_V\in End_k(V) $$ is nilpotent ($\pi$ is the representation morphism).

Groupes et algèbres de Lie(Lie groups and Lie algebras), beginning of Chap VII. See esp. Prop 9 in §1.3 (on decomposition of modules over nilpotent Lie algebras). $\endgroup$ – YCor Mar 17 '18 at 8:38