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 $\mathfrak{q} \subset \mathfrak{g}$?




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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,](https://math.stackexchange.com/questions/4172661/does-every-nilpotent-element-in-a-real-lie-algebra-lie-in-some-minimal-parabolic) for which I believe I have found an argument now, but I yet need to formalize it).



Complexifications of minimal parabolic subalgebras
NB: for brevity I'm going to refer to Borel subalgebras and parabolic subalgebra as Borels and parabolics, respectively


Even for a split (or quasi-split) Lie algebra not all complex Borels are the complexification of real Borels. Otherwise, for example, the real and complex flag manifolds would be identical. Naturally, a Borel cannot contain another one since they are supposed to be maximal solvable/minimal parabolic. So for these cases the condition would appear to be necessary as well as sufficient.


What is true is that a real parabolic $ \mathfrak{p} \leq \mathfrak{g} $ complexifies to a complex parabolic $ \mathfrak{p}^{\mathbb{C}} \leq \mathfrak{g}^{\mathbb{C}} $. Such a complex parabolic contains a family of Borels (of $\mathfrak{g}^{\mathbb{C}}$) and any one of these determines $ \mathfrak{p}^{\mathbb{C}} $ and thus $\mathfrak{p}$ uniquely.
To be more precise parabolics come in a number of conjugacy classes in correspondence with subsets of the Dynkin diagram and a given Borel subalgebra defines a unique parabolic in each conjugacy class that contains it. In particular we get a unique one in the class which contains the complexifications of the minimal parabolics in our real form. Because we chose it to be a complexification we can recover its real form but that wouldn't work otherwise. The uniqueness we mentioned above tells us that if a Borel $\mathfrak{b}$ is contained in some complex parabolic $\mathfrak{q}$ which is conjugate to $\mathfrak{p}^{\mathbb{C}}$ but is not a complexified parabolic itself then $\mathfrak{b}$ is not contained in any complexified minimal parabolic.


So now we have to ask if all Borel subalgebras contained within a given $ \mathfrak{p}^{\mathbb{C}} $ are of the form described by Knapp. I couldn't work that out off the top of my head but certainly there is some necessary condition.


