Remark 2

Let $\mathfrak{g}\subset \mathfrak{g}\mathfrak{l}$(V), $\mathfrak{g}'\subset \mathfrak{g}\mathfrak{l}(V')$ be decomposable Lie algebras, $\varphi$ an isomorphism from $\mathfrak{g}$ to $\mathfrak{g}'$. Note that $\varphi$ does not necessarily transform semi-simple (resp. nilpotent) elements of $\mathfrak{g}$ to semi-simple (resp. nilpotent) elements of $\mathfrak{g}'$ (Exerc. 2). However, this is the case if $\mathfrak{g}$ is semi-simple (Chap. I, §6, no. 3, Th. 3).