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.

soluble. (It is a pretty safe bet that, if checking whether a property of Lie groups can be deduced from an analogous property of Lie algebras in positive characteristic, then the answer is probably 'no'. The proper definition of 'Borel subalgebra' here is 'algebra of Borel'.) $\endgroup$beany Borel subgroup, or Borel subalgebra, over a non-algebraically closed field. $\endgroup$2more comments