so let's consider for example the root polytope of a3, and try to interpret its facial structure as embodying the incidence geometry in the coadjoint partial flag picture...

so the root polytope is a "cuboctahedron"...

hmm, but wait a minute... do we really know how to interpret the facial structure in this case??

coadjoint partial flags appear as vertexes...

??edges of two colors??

i'm confused...

??a point corresponds to a 2d space of coadjoint partial flags... namely those where the point is the point, and the hyperplane is any hyperplane through it... of which there's a 2d space... dual to 2d space of points on a hyperplane...

a line corresponds to a 3d space of coadjoint partial flags??? i'm not getting that... seems more like 2d ... ????....