TITLE: Relationship between parabolic subgroups and parabolic subalgebras over non-algebraically-closed fields
QUESTION [5 upvotes]: Let $\mathbb{K}$ be an arbitrary field, and $G$ an algebraic group (or group variety?) over this field. A Borel subgroup of $G$ is a connected solvable subgroup variety $B$ of $G$ such that $G/B$ is complete. A parabolic subgroup of $G$ is a subgroup variety $P$ such that $G/P$ is complete.
From section 29.4 of Tauvel and Yu, I know that if $\mathbb{K}$ is algebraically closed and has characteristic zero, then a Borel subalgebra of a finite-dimensional Lie $\mathbb{K}$-algebra $\mathfrak{g}$ (not necessarily semisimple) is defined to be a maximal solvable subalgebra $\mathfrak{b} \leq \mathfrak{g}$. A parabolic subalgebra $\mathfrak{p} \leq \mathfrak{g}$ is any subalgebra containing a Borel subalgebra. 
Tauvel and Yu prove that under these hypotheses on the field ($\mathbb{K}$ is algebraically closed and has characteristic zero), if $G$ is a connected algebraic group with Lie algebra $\mathfrak{g}$, then parabolic subalgebras of $\mathfrak{g}$ are precisely the Lie algebras of parabolic subgroups of $G$. 
Now if we generalize so that $\mathbb{K}$ is not algebraically closed (or more generally, but less importantly, when the characteristic is nonzero), I would like to know to what extent these definitions and correspondences carry over? Moreover, if $\overline{\mathbb{K}}$ is the algebraic closure of $\mathbb{K}$, what properties are preserved and reflected by "extending" the scalars?
For my particular purposes, I am most interested in the case $\mathbb{K}=\mathbb{R}$. However, it seems that everywhere I look the complex case is emphasized. I like to think very categorically, and in a "top-down" sort of way, so I am hoping to get a good overview of where I am going and what to expect before I slog through the complex theory.
If it helps at all, I trying to get a handle on parabolic subgroups and subalgebras so that I can reach a more comprehensive understanding of parabolic Cartan geometries and (eventually) BGG resolutions. 
Unfortunately for me, my knowledge of algebraic geometry and algebraic groups is very limited, so I apologize if the answer to my question is very obvious.

REPLY [4 votes]: I think the best book for you is Parabolic Geometries I : Background and General Theory by Andreas Čap and Jan Slovák. The first half covers the basics of Cartan geometries and classification of real Lie groups. The parabolic Cartan geometries are the content of its second half. As far as I know, there is no book for BGG resolutions as of yet. My feeling is that we will have to wait a few years for new results "to stabilize". E.g. for the most advanced stuff in this direction you can checkout arXiv:1510.03331 and its sequel. 
Some answers to your questions can be found in the third chapter of the aforementioned book. The main point here is that from the point of view of parabolic geometries you need to care only about subgroups of $G$ that preserve certain filtrations.