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).*
> 
> 
> 



About indecomposability and nilpotence
The answer is positive. Details and link to Bourbaki, Lie groups and Lie algebras, Chap VII, are given in [the answer here](https://math.stackexchange.com/a/2694748/35400).


