abstract:In the mathematical field of Lie theory, a split Lie algebra is a pair (\mathfrak{g}, \mathfrak{h}) where \mathfrak{g} is a Lie algebra and \mathfrak{h} < \mathfrak{g} is a splitting Cartan subalgebra, where "splitting" means that for all x \in \mathfrak{h}, \operatorname{ad}_{\mathfrak{g}} x is triangularizable. If a Lie algebra admits a splitting, it is called a splittable Lie algebra.