Question regarding the relationship of solvability of Lie algebras and short exact sequences.

Question

This is about the fact that for a short exact sequence of Lie algebras $$0\rightarrow K\rightarrow L\rightarrow Q\rightarrow 0$$ we have that L is sovable iff K and Q are solvable. In essence i would like to know why the following argument doesent work: Consider the s.e.s. $$0\rightarrow K\xrightarrow[]{\text{f}} L\xrightarrow[]{g} Q\rightarrow 0$$ with K and Q nilpotent. As Q is nilpotent, we find an m s.t. $C^mQ=0$. Then $$g(C^mL)=C^mg(L)=C^mQ=0$$ so $C^mL\subset im(f)$. As the image of f is nilpotent we find an n , s.t. $C^n im(f)=0$. Thus $C^{n+m} L=0$. But my professor stated that this is wrong, in the sense that only solvable Lie algebras have this property but in general nilpotent Lie algebras dont. Can somebody spot the Error and maybe give a counter example? Thanks!