I need a hint on how to do one problem in "Introduction to Lie Algebras and Representation Theory" by James E. Humphreys.
Suppose that $L$ is a reductive Lie algebra ($\textrm{Rad}\space L=Z(L)$) over a field $F$ (assume it is algebraically closed and char $F=0$).Show that all finite dimensional representations of $L$ in which $Z(L)$ is represented by semisimple endomorphisms are completely reducible.
This is my attempt so far:
Assume that $\phi:L\rightarrow gl(V)$ be a finite dimensional representation of $L$ that satisfies the property mentioned above. It is easy to check that $\phi(Z(L))$ is a family of commuting endormorphisms on $V$. Since each element in $\phi(Z(L))$ is diagonalisable, we can decompose $V$ as a direct sums of $V_{\alpha}=\{v\in V|\phi(y)v=\alpha(y)v,y\in Z(L)\}$, where $\alpha\in L^*$, the dual space of $L$. It is easy to see that each $V_{\alpha}$ is an $L$ submodule of $V$ (this follows from invariance lemma).
Now I only need to show that each $V_{\alpha}$ is irreducible. However, I am not sure how to do that. Perhaps anyone can give hints (preferably not the full solution).