$F_4$ flag variety As flag variety or a homogeneous variety is a quotient $\Sigma=G/P$ of a reductive Lie group $G$ by one of its parabolic subgroups $P$. The subgroup $P$ fixes a flag of subspaces of standard representation $V$ of $G$. There is an embedding of projective varieties $\Sigma\subset \mathbb P V_{\lambda}$, where $V_{\lambda}$ is some highest weight representation of $G$.  
For the exceptional Lie group of type $G_2$, if we consider its highest weight representation for highest weight $\omega_2$ then we have an embedding of a homogeneous variety  $\Sigma\subset \mathbb P V_{\omega_2}$. Since its a subvariety of Gr(2,7), which can easily be seen to be a "flag variety", so we can some how realise  this $G_2$ variety as a flag variety. 
If we consider an exceptional Lie group $F_4$ and take its highest weight representation with highest weight $\omega_1$ then $V_{\omega_1}$ is 26 dimensional and we have an embedding of $F_4$ homogeneous variety $\Sigma ^{15}\subset \mathbb P V_{\omega_1}$, which is a codimension 10 embedding. My question is that how can we realise this variety as a "flag variety" or is it also a subvariety of some other standard flag variety? 
 A: Section 9.1 of Carr-Garibaldi contains a nice explanation of this:
http://arxiv.org/abs/math/0503201
In general, this paper explains how to realize G/P as flags of special kinds of subspaces for any type (except $E_8$)
A: Boris Rosenfelds book "The Geometry of Lie Groups" seems to address that question in Thm. 7.35 (page 358). Be warned that Rosenfelds book is both fascinating (for the wealth of its knowldege) and very frustrating (for reasons that you can find out for yourself).
If I understand these matters correctly (a non-negligible "if"), there are notions of "points", "lines", "planes" and "symplecta" as submanifolds of the Cayley plane $\mathbb{O}P^2$. These are submanifolds of $\mathbb{O} P^2$ that are isometric to $\mathbb{C}P^1$, $\mathbb{C}P^2$, $\mathbb{C}Sp^5$, resp. the "absolute hermitian conic of $\mathbb{O}P^2$" (whatever that may be). The parabolic quotients $F_4/P$ then represent flags of these structures, with incidence requirements. (I think this approach goes back to Freudenthal, so you might find a clearer description in his writings.) 
A: As far as I remember $F_4/P_{\omega_1}$ is a hyperplane section of $E_6/P_{\omega_1}$.
