Exercise 2, § 5

Let $u$ (resp. $v$) be a non-zero semi-simple (resp. nilpotent) endomorphism of V. Then the map $\lambda u \rightarrow \lambda v(\lambda \in k)$ is an isomorphism from $\mathfrak{g}=ku$ to $\mathfrak{g}'=kv$ which does not take semi-simple elements to semi-simple elements.