If $G$ is a *classical* semisimple algebraic/Lie group over an algebraically closed field (maybe just say $\mathbb{C}$), viꝫ. $\mathit{SL}_n$, $\mathit{SO}_n$, $\mathit{Sp}_n$ (isogenies irrelevant here), there is a simple description of the quotients $G/P$ by parabolic subgroups of $G$ as grassmannians or partial flag varieties:

$\mathit{SL}_n/P$ parametrizes flags $V_1 \subseteq \cdots \subseteq V_r$ of vector subspaces of fixed specified dimensions $0 < d_1 < \dotsb < d_r < n$ in a fixed vector space of dimension $n$;

$\mathit{SO}_n/P$ parametrizes flags $V_1 \subseteq \dotsb \subseteq V_r$ of totally isotropic vector subspaces of fixed specified dimensions $0 < d_1 < \dotsb < d_r \leq \frac{n}{2}$ in a fixed vector space of dimension $n$ endowed with a nondegenerate quadratic form, except that when $d_r = \frac{n}{2}$ (for $n$ even, of course), we can also impose that $V_r$ belongs to one or the other of the two families of $(n/2)$-dimensional subspaces;

$\mathit{Sp}_n/P$ parametrizes flags $V_1 \subseteq \cdots \subseteq V_r$ of totally isotropic vector subspaces of fixed specified dimensions $0 < d_1 < \dotsb < d_r \leq n$ in a fixed vector space of dimension $2n$ endowed with a nondegenerate alternating form.

(Of course, the above can be more clearly worded to specify the $(d_i)$ in function of the nodes chosen in the Dynkin diagram of $G$ to define $P$.)

**Question:** Is there a similar description, when $G$ is an *exceptional* semisimple group, of its parabolic quotients $G/P$ as parametrizing subspaces or partial flags of some space endowed with additional structure on which parts of that structure are “isotropic” in a certain sense?

The kind of answer I am looking for is something like this:

- $G_2/P$ parametrizes $1$-dimensional, or $2$-dimensional, or nested $V_1\subseteq V_2$ (with $\dim V_i = i$) subspaces of the $7$-dimensional purely imaginary part $A^0$ of the octonion algebra $A$ (meaning split, as we are over an algebraically closed field) such that the octonion product is identically zero on the subspace

(I think this is correct, but I don't really have a reference clearly stating this.)

What about $F_4$, $E_6$, $E_7$, $E_8$? (Of course, this all depends on appropriate representations having been constructed. For example, I expect the descriptions of $E_8/P$ to be something like parametrizing subspaces or partial flags in the Lie algebra $\mathfrak{e}_8$ since the adjoint representation generates every other fundamental representation by taking alternating powers. But what kind of subspaces, exactly?)

Such a description doesn't seem to be in J. F. Adams's *Lectures on Exceptional Groups*. I think B. Rosenfeld's *Geometry of Lie Groups* §7.6 may be supposed to answer the question (under the name “fundamental figures” and “parabolic figures”), but it is extremely difficult to decipher (and the fact that it treats the real, not-necessarily-split, case, of course makes it even more difficult). An answer might also be in Landsberg & Manivel's paper “The Geometry of Freudenthal's Magic Square”, but I was unable to find where exactly (and even then, it doesn't seem to be couched in the language of subspaces and flags).