Let $\mathfrak{g}$ be a real semisimple Lie algebra with complexification $\mathfrak{g}_\mathbb{C}$. Recall that a parabolic subalgebra in $\mathfrak{g}_\mathbb{C}$ is one which contains a Borel subalgebra, and a parabolic subalgebra in $\mathfrak{g}$ is one which complexifies to a parabolic one in $\mathfrak{g}_\mathbb{C}$.
Question: Given a Borel subalgebra $\mathfrak{b} \subset \mathfrak{g}_\mathbb{C}$, is it contained in the complexification $\mathfrak{q}_\mathbb{C}$ of some minimal parabolic subalgebra $\mathfrak{q} \subset \mathfrak{g}$?
Problem VII.13 in Knapp's "Lie Groups beyond an introduction" says this is true, under the additional condition that $\mathfrak{b}$ is built from the complexification of a maximally noncompact Cartan subalgebra of $\mathfrak{g}$, in the usual way via a root space decomposition and choice of positive roots. I am wondering to what extent we can drop this requirement -- or if this is a strong requirement at all?
Note that Knapp defines minimal parabolic subalgebras differently than I do here, but I believe their notion should be equivalent (this is related to the first part of my question here on math.stackexchange, for which I believe I have found an argument now, but I yet need to formalize it).