Exercise 11, § 3

Let $\mathfrak{a}=\prod\mathfrak{a}_i$ be a finite product of semi-simple Lie algebras. A subalgebra $\mathfrak{q}$ of $\mathfrak{a}$ is a parabolic (resp. Borel) subalgebra if and only if it is of the form $\mathfrak{q}=\prod\mathfrak{q}_i$ where, for all $i,\mathfrak{q}_i$ is a parabolic (resp. Borel) subalgebra of $\mathfrak{a}_i$.