?? so is a2 in b2 ???? ..... ?? via "axis configuration" variety being projective plane ?? .... ?? question sounds vaguely related to some root system idea we had ... ??? pattern involving rank 2 systems .... talking with john huerta ... idea didn't go far; i almost sort of remember what happened to it ... ??? .... might be able to dig up in notes .... ???? ...

??? a2 borel ... ??? .....

?? no wait, so a2 obviously _is_ in b2 ??? .... no no no .... ok, some silly confusions here ... so(3) vs sl(3) .... ??? .... b2 as so(2,3) so includes so(3) .... ???? ....

?? so in trying to line up a2, b2, g2 in some sort of attempted pattern where each involves something moving on a 2-sphere... ?? resp s^0, s^1, s^0 X s^2 ... ???? ..... ???? 2-sphere always has its so(3) metric symmetry .... ???? .... ??? but that makes it seem like it's not just straightforwardly corresponding to one of the two simple roots .... which would give so(2,1)'s, right ?? .... ??? ....

... hmmm ...

... maximal compact .... ???? ....

?? the thing moving on the 2-sphere as always having non-singular "one-move-away" schubert variety .... ??? making it seem like there _is_ some nice simple long root / short root / invariat distribution / root system pattern going on here .... ???....

?? 2-sphere versus thing moving on it, vs stopped configurations vs paths of motion ... ???? ....

?? would be nice to find notes about that (...) discussion with huerta ... at del taco ...