Take the generalized flag variety $SO(2n,\mathbb{C})/B$, considered as the moduli of isotropic flags (according to the form $\langle e_i, e_{2n+1-j}\rangle=\delta_{ij}$) $$F_1\subset F_2\subset\cdots F_n\subset\mathbb{C}^{2n}$$ with $\dim(F_i)=i$ and $\dim(F_n\cap\mathrm{Span}(e_1,\ldots,e_n))$ even.

Let $P$ be the maximal parabolic subgroup of $SO(2n)$ omitting one of the two nodes at the forked end. Then $G/P$ is the moduli of isotropic subspaces of dimension $n$ in $\mathbb{C}^{2n}$.

For one of those end nodes, the description of the projection $G/B\rightarrow G/P$ is obvious; one just sends the point corresponding to the full isotropic flag to the point corresponding to the maximal subspace $F_n$.

In explicit linear algebra terms, what is the projection $G/B\rightarrow G/P$ for the maximal parabolic corresponding to the other end node on the forked end? In other words, what is the other way to get a maximal isotropic subspace out of a full isotropic flag?

A giambelli formula for even orthogonal Grassmannians. An explicit example in D4 is the pair 34127856 and 46718235 (usual embedding into S8), which are incomparable in Bruhat order but comparable using dimensions of subspaces. $\endgroup$ – Alexander Woo Jul 17 '16 at 4:02