The inverse of Weyl's Theorem

Question

$L$ is a Lie algebra. If any finite dimensional repersentation of $L$ is completely reducible, then $L$ is semisimple. I have already proved that $L$ is reductive, i.e. $\text{Rad}(L)=Z(L)$, by considering the adjoint representation. However, I stucked here and did not know what to do.

Answer 1

What is left to show is $Z(L)=0$, right? Now $Z(L)$ is abelian; if it is $\neq 0$, can you come up with a representation of it which is not completely reducible (hint: upper triangular matrix)? Then lift that to a representation of $L$.