?? 4d distribution on g2 config space ??????? ..... ????? ....... i mean, mystery (?? ...) of why there doesn't seem to be one ... ??? .... ?? so(3,4) .... ???? relationship between "2 rolls away" for g2 and certain orientation for so(3,4) .... ???? ..... ??? perp of 1d null subspace ... conceptual / geometric interpretation thereof ..... ???? .....

?? ok, so seems real likely that the 2-rolls-away subalgs in the split octonions are in fact _exterior algs_ rather than split quaternion .... ?? should have been pretty obvious, and huerta and i probably once talked about it pretty long ago .... ?? but i _did_ try checking last week or so the anticommutator on my canonical 2-rolls-away algebra and it seemed non-tirivial ... which now seems like must be a mistake; i'll try checking up on that check ....

ok, here's where i tried the check, quoting from here:

?? meanwhile, anticommutator of (i,1) and (j,1) .... ??? as (k+1,i-j) + (-k+1,j-i) = (2,0), so that's a positive constant, so good representatives a,b should be (i,1),(-j,-1) .... ???? .... ?? so then a+b is (i-j,0) ..... ????? ..... hmmmmm ...... ????

?? so .... ?? can we find our mistake here ??? .... hmm, i _still_ seem to be getting (2,0) here, using wikipedia convention .... ?? .... ?? and even using other convention ?? ...

hmm, huerta's email might give some explanation of the glitch here ... ?? involving tweak map ??? ..... ????? ..... ???? .....


?? anyway, lots of stuff threatening to make sense here ... including connections with so(3,4) and tweak map ..... ???? .....

?? split octonion product in 2-rolls-away / exterior algebra case as giving the (?? unique ...) pivot point witnessing the 2 rolls ..... ???? ....

?? so(3,4) .... 5d space of 1d isotropics ("projective light cone" ...), 7d space of 2d, 6d space of 3d .... ???? some of all those showing up here in interesting ways .... ????? ......

?? does "graph of isometry" property get screwed up in complex case ???? ...... ???? ..... ???? biquaternions as not division algebra ..... ???????? ....... not sure i said the (??? ....) right thing here ..... ?????? .... ?? some correction (?? ...) huerta pointed out to me at some point .... ????? .... ...... ?? stuff we thought about in connection with homogeneous coordinate alg of certain flag varieties around october 2011 ... trying to prepare for talk which as it turned out didn't go much in that direction ... ??? maybe around here ???? .... isometry as invertible ..... ???????? ........ ?? flag variety that (?? over the reals ??? ...) mimics group in way but is actually far from being alg gp ...... ??????? ...... ????? ...

?? exterior subalg of split octonions .... ??? .... 2-roll path vs 1-non-roll path ... so(3,4) ... ??? .....