Definition 1

A subalgebra of $\mathfrak{g}$ containing $\mathfrak{h}$ and satisfying the equivalent condition in Prop. 7 is called a Borel subalgebra of $(\mathfrak{g},\mathfrak{h})$.

A subalgebra $\mathfrak{b}$ of a splittable algebra $\mathfrak{g}$ is called a Borel subalgebra of $\mathfrak{g}$ if there exists a splitting Cartan subalgebra $\mathfrak{h}'$ of $\mathfrak{g}$ such that $\mathfrak{b}$ is a Borel subalgebra of $(\mathfrak{g},\mathfrak{h}')$.

Let $(\mathfrak{g},\mathfrak{h})$ be a split reductive Lie algebra. Let $\mathfrak{g}=\mathfrak{c}\times \mathfrak{s}$ with $\mathfrak{c}$ commutative and $\mathfrak{s}$ semi-simple. A subalgebra of $\mathfrak{g}$ of the form $\mathfrak{c}\times \mathfrak{b}$, where $\mathfrak{b}$ is a Borel subalgebra of $(\mathfrak{s},\mathfrak{h}\cap \mathfrak{s})$, is called a Borel subalgebra of $(\mathfrak{g},\mathfrak{h})$.

With the notations of Prop. 7, we also say that $\mathfrak{b}$ is the Borel subalgebra of $\mathfrak{g}$ defined by $\mathfrak{h}$ and C (or by $\mathfrak{h}$ and the basis of R associated to C).