Definition 3

Let $\mathfrak{a}$ be a semi-simple (or more generally reductive) Lie algebra and $\overline{k}$ an algebraic closure of $k$. A subalgebra $\mathfrak{m}$ of $\mathfrak{a}$ is said to be parabolic (resp. Borel ) if $\mathfrak{m}\otimes_k\overline{k}$ is a parabolic (resp. Borel ) subalgebra of $\mathfrak{a}\otimes_k\overline{k}$.

If $\mathfrak{a}$ is splittable, Prop. 14 shows that this definition is equivalent to Definition 2 (resp. to Definition 1).