Remark 1

The semi-simplicity assumption on $\pi$ is equivalent to saying that $\pi (x)$ is semi-simple for all $x\in \mathfrak{c}$. Note that this assumption is satisfied when $\pi$ is obtained from the identity representation $\mathfrak{g}\rightarrow \mathfrak{g}\mathfrak{l}(V)$ by the successive application of the following operations: tensor product, passage to the dual, to a subrepresentation, to a quotient, to a direct sum.