so i think that part of what's going on here is that among the invariant distributions the integrable ones have a very basic special place, and it's important to work out how they fit in. and this to some extent justifies and motivates focusing on the total flag variety and treating the partial ones as just arising from the integrable invariant distributions, though i'd still like to clarify some more the way (probably perfect sensible once you understand what's going on, unlike me so far) in which _not_ quite invariant distributions seem to be entering here.

from a certain viewpoint perhaps an integrable invariant distribution corresponds to "a schubert variety whose logical intension uses only the equality predicate" (or something like that... or from a slightly different viewpoint, only equivalence predicates...). a leaf of the foliation that you get by integrating the distribution can be thought of as a "residual partial flag" (or something like that... ??or maybe i mean a "residual geometry" ??? .... try to straighten this out...)... ??and can also be thought of as a schubert variety?? ...

back to the a2 example:

ab=
cde
=fg

borel subalgebra abcd, down-sets of g>e,f = [],[e],[f],[ef],[efg]

among these [],[e],[f],[efg] are integrable... ??