Corollary 3

Let $\mathfrak{h}$ be a Cartan subalgebra of a reductive Lie algebra $\mathfrak{g}$.

a$)\mathfrak{h}$ is commutative.

b) If $\rho$ is a finite dimensional semi-simple representation of $\mathfrak{g}$, the elements of $\rho (\mathfrak{h})$ are semi-simple.

Let $\mathfrak{c}$ be the centre of $\mathfrak{g}$, and $\mathfrak{s}$ its derived algebra. Then $\mathfrak{g}=\mathfrak{c}\times \mathfrak{s}$, so $\mathfrak{h}=\mathfrak{c}\times \mathfrak{h}'$, where $\mathfrak{h}'$ is a Cartan subalgebra of $\mathfrak{s}$ (Prop. 2). In view of Th. $2,\mathfrak{h}'$ is commutative, hence so is $\mathfrak{h}$. Moreover, $\rho (\mathfrak{h}')$ consists of semi-simple elements and so does $\rho (\mathfrak{c})$ (Chap. I, §6, no. 5, Th. 4); assertion b) follows.