Can the intersection of a maximal parabolic with a closed sub-group contain more than one maximal parabolic?

Question

Suppose that we have a closed embedding $G_1\hookrightarrow G_2$ of reductive groups (say over $\mathbb{Q}$), and suppose that we have a maximal parabolic sub-group $P_2\subset G_2$, and a minimal parabolic $P_1\subset G_1$. Is it possible to have two different maximal parabolic sub-groups of $G_1$ contained in $P_2$ and containing $P_1$?

Actually, is it even possible that there are two different maximal parabolics of $G_1$ contained in $P_2$?

Even more optimistically, if there is a maximal parabolic sub-group of $G_1$ contained in $G_1\cap P_2$, does that make $G_1\cap P_2$ a parabolic sub-group and thus equal to the maximal parabolic it contains?

EDIT: As Jim Humphreys points out below, there is one pathological case, where $G_1\cap P_2$ might be all of $G_1$. In which case, all parabolic sub-groups of $G_1$ will be contained in $P_2$! This is the case, for example, when $G_1$ is a Levi sub-group of $G_2$. But Angelo's answer below shows that, excluding this possibility, the answer to my third question is 'yes'.

Answer 1

If $P$ is a parabolic subgroup of a reductive group $G$ and $H$ is a closed subgroup of $G$ containing $P$, then $G/H$ is a quotient of $G/P$, so it is projective, and $H$ is parabolic. It follows that a maximal parabolic subgroup is maximal among all proper closed subgroups, which would seem to imply that the answer to all of your questions is positive.

Answer 2

There are several questions being asked (and an unexplained reference to a field of definition), but the answer to at least one of them is no: Take $G_1 = G_2 = \mathrm{SL}_3(\mathbb{C})$, with a given minimal = maximal parabolic subgroup involving a single simple root subgroup relative to some choice of positive roots; this parabolic clearly won't lie in two distinct maximal ones. (I'm assuming "minimal" excludes a Borel subgroup and "maximal" means proper, though a reductive group might be just a torus.)

So it's a good idea to separate out the more precise question you have in mind and specify whether a field of definition is really involved for the various groups and subgroups involved.

P.S. I've only pointed to the trivial case where there is just one reductive group. But I think your line of questioning will also run into obstacles in situations where the group and subgroup share a common maximal torus but the subgroup is of "pseudo-Levi" type. An example with both $G_1, G_2$ simple of rank 2 occurs when you consider the reductive subgroup of a group of Lie type $G_2$ (in root system notation!) which involves the long roots and has Lie type $A_2$. These pseudo-Levi subgroups are described in terms of proper subsets of vertices in the extended Dynkin diagram which fail to define Levi subgroups of parabolics.

To summarize, there are pairs $G_1 \subset G_2$ and respective parabolic subgroups (all defined over a given subfield of $\mathbb{C}$ such as $\mathbb{Q}$) for which the answers to the various questions asked can be either yes or no. Another example: Take a reductive subgroup $G_1 \cong \mathrm{SL}_2(\mathbb{C})$ of $G_2 = \mathrm{SL}_3(\mathbb{C})$ with a Borel subgroup $P_1$ (involving one simple root $\alpha$) as minimal = maximal parabolic of $G_1$; then $P_1$ and the opposite Borel both lie in the standard maximal parabolic $P_2$ of $G_2$ having $-\alpha$ as a root.