so as a potential guide to the g2 case (as well as probably lots of other reasons), let's try thinking a bit about the long root subalgebra so(2,2) of b2 = so(2,3).

so let's take a highest weight vector of the adjoint representation of so(2,3) and consider it's orbit under so(2,3), but then let's consider how that breaks up into orbits under the long root subalgebra so(2,2).

(by the way does the inclusion of g2 into so(3,4) "induce" an inclusion of the long root subalgebra sl(3) of g2 into the one so(3,3) of so(3,4)? or something? doesn't seem particularly likely, even though there _is_ a nice inclusion of sl(3) into sl(4) = so(3,3)... ??...)

so suppose that we have an orientation-preserving linear isometry x from e^2 to e^2, and a 1d subspace s of the domain e^2. then this gives us also a unique orientation-reversing linear isometry y that agrees with x on s.

hmm, so consider the hoop configurations that confine the hoop to the favorite equator, and the "lines" whose "axis" lies on that equator. this seems to correspond nicely to the so(2,2) geometry inside the so(2,3) one.

are there any elements of so(2,3) that don't preserve the equivalence relation on flags given by "share the same special antipodal pair of points on the material hoop", and if so then can we nicely visualize any of them?? are they "boost-like" or something?? hmm, is this equivalence relation maybe "generalized simultaneity" or something?? i think that it probably is; the material hoop plays the role of "time". so does it make sense to "boost by an isometry from time to space" or something??

hmm, so let's try a naively analogous approach with g2. let's try taking the "favorite equator" of the unit-length quaternions to be the imaginary ones. so is it true that the "second quaternion components" of the 6 projective octonions in the favorite g2 apartment are imaginary?? or something?? i think that it is...

??this seems to be fitting with the idea that the a2 total flag variety "has no interior coordinates". or something...

??are we getting the wrong number of constraints on the configurations here??