Remark 5

Let $\mathfrak{g}$ be a reductive Lie algebra. Then $\mathfrak{g}=\mathfrak{c}\times \mathfrak{s}$ where $\mathfrak{c}$ is the centre of $\mathfrak{g}$ and $\mathfrak{s}=\mathscr{D}\mathfrak{g}$ is semi-simple. The Cartan subalgebras of $\mathfrak{g}$ are the subalgebras of the form $\mathfrak{h}=\mathfrak{c}\times \mathfrak{h}'$ where $\mathfrak{h}'$ is a Cartan subalgebra of $\mathfrak{s}$ (Chap. VII, §2, no. 1, Prop. 2). Then $\mathfrak{h}$ is called splitting if $\mathfrak{h}'$ is splitting relative to $\mathfrak{s}$. This leads in an obvious way to the definition of splittable or split reductive algebras.