Definition 2

A subalgebra of $\mathfrak{g}$ containing $\mathfrak{h}$ and satisfying the equivalent conditions of Prop. 11 is called a parabolic subalgebra of $(\mathfrak{g},\mathfrak{h})$. A parabolic subalgebra of $\mathfrak{g}$ is a parabolic subalgebra of $(\mathfrak{g},\mathfrak{h}')$ where $\mathfrak{h}'$ is a splitting Cartan subalgebra of $\mathfrak{g}$.

This definition extends immediately to the case in which $(\mathfrak{g},\mathfrak{h})$ is a split reductive Lie algebra.