Definition 1

Let $\mathfrak{g}$ be a semi-simple Lie algebra. A Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ is called splitting if, for all $x\in \mathfrak{h}$, ad$_{\mathfrak{g}}x$ is triangularizable. A semi-simple Lie algebra is called splittable if it has a splitting Cartan subalgebra. A split semi-simple Lie algebra is a pair $(\mathfrak{g},\mathfrak{h})$ where $\mathfrak{g}$ is a semi-simple Lie algebra and $\mathfrak{h}$ is a splitting Cartan subalgebra of $\mathfrak{g}$.