?? so consider the subalgebra of the split octonions generated by the nilpotents (i,1) and (j,1) ...

?? seems like it's probably a split quaternion subalgebra ... ?? but perhaps essentially different in nature from the more standard (?? ...) one generated by (i,1) and (i,-1) ???....

?? to what extent does the "tweak map" screw up ideas about ... ??? certain hopefully nice conceptually simple relationships of g2 geometry to so(7) and so(8) and so forth .... ???? ..... ?? maybe not that much ?? ... ?? maybe all (?? ...) that stuff lies comfortably in the background, against which the g2 lines are carved out (?? in a funny way connected with the annoyance of the tweak map ?? ...) as the foreground .... ???? ...

?? a-b .... ??? ... ?? vs a+b ?? ......

?? confusion here about 2 ray halves of line ??? ?? hmm, maybe it's pretty obvious that as soon as both halves of the signature go up to 2 then the cone doesn't break cleanly in two pieces .... ?? ...

?? some sort of funny game involving relating discriminant of binary quadratic form to determinant of traceless operator whose matrix is related by "exchanging the main and anti-main diagonals" ?? .... ????? .....

a b
b c

b -a
c -b

???? ..... ???

ac-bb ??? .... ???? ....

?? "checkerboard" aspect here ?? ... ?? allowing b to be half-integer ??? ... ??? .... ?? gauss vs someone else ??? ...... ?? jordan product here ?????? ..... ????? .... ??? jordan vs lie here ??????? ....... ??? clifford-jordan algebra and traceful operators .... ???? ....

?? clifford-jordan algebra of binary quadratic forms wrt discriminant, vs underlying jordan algebra of associative algebra of 2-by-2 matrixes .... ???? ..... ?? "cone" .... ???? ... "positive" / "future" .... "light" ... ??? ... "sky" .... ????? ....

b -a
c -b

e -d
f -e

be-cd -ae+bd
bf-ce -af+be

eb-fa -db+ea
ec-fb -dc+eb

be-cd+eb-fa 0
0 -af+be-dc+eb

?? that does seem to confirm it by gruesome calculation ... ?? that the clifford-jordan product of binary quadratic forms is essentially the same as the anticommutator of traceless operators on k^2 .... ???? .....

?? and determinant of traceless operators on k^2 is the "quadratic (?? de- ??)polarization" of the bilinear operation "anticommutator" ??? .... ???? .... not sure i said that right ... or in what ways it can/should be generalized .... ??? ...

??? hmm, i was going to ask about analog for unsplit integral quaternions of what we're doing here with split integral quaternions .... ?? but then perhaps this opens a can of pleasant worms ... ?? rather than just split and unsplit quaternions in the integral case, maybe there's a whole interesting family of them ... corresponding to different "discriminants" ... ???????? ...... ?????? .....

?? underlying jordan algebra of d-integral quaternions ..... ???? .... ?? "d-discriminant of binary quadratic form" ???? ...... ?????? ....

?? checkerboard aspect here ... ??? interesting lattices in the quaternions .... ???? ... conway .... ??? ....

?? ideas about "discriminant of binary quadratic form as universal such form, almost sort of ..." .... ???? .... ?? connections here ??? ..... ???? how did that (?? ...) go .... ??? "genus zero analog of taniyama conjecture" .... ????? ..... ???? ..... ???? ....

?? "spin factor" stuff ... ?? generalizing some stuff above ?? ... ??? did i see huerta do stuff like that once ??? ..... jordan algebra .... ???? ..... ?? hmm, arbitrary "spin factor" vs one associated with ..... ????? division algebra ???? .....

??? "berezinian" ..... ???? ..... ????? .....

?? "d-integral quaternions" for discriminant value d vs "q-integral quaternions" for quadratic form q .... ????? ....

??? "witt ..." .... ???? ....

?? trying to get back to the octonions here .... ?? ...

?? "positive indefinite" binary quadratic form .... ??

p 0
0 0

0 0
0 q

?? corresponding traceless operators .... ??? ....

0 -p
0 0

0 0
q 0

?? ..... anticommutator of traceless operators on k^2 ...?? ... ??as "polarization bilinear form of determinant quadratic form" .... ???? ....

?? hmmm, seem to be getting anticommutator here as _negative_ ..... ???? .... (is this due to some annoying convention mismatch somewhere ?? ... ??? ...)

?? ab+ba and bc+cb > 0 => ac+ca > 0 ??????? .... ?? maybe not quite right way to say it ... ??? .... ?? mixed vs indefinite confusion here ??? ...

?? anticommutator of (i,1) and (i,-1) vs of (i,1) and (j,1) ... ??? ....

?? once that's straightened out ... ?? that is, scaling so the anticommutators are negative .... ???? .... ?? then try ... ?? conjugating by exponential of (a+b)*k ... ??? ....

?? "alternative" = "2-at-a-time associative" ?? ... ??? so don't have to worry about associativity here ??? ....

anticommutator of (i,1) and (i,-1) as .... ??? (-2,-2i) + (-2,2i) = (-2,0) ... so that's a negative constant, so this seems to be a "good" pair a,b of representatives .... so then a+b = (2i,0) .... ??? ... ??? then conjugating by exponentials of this ..... ??? ... (r+si,0) with r^2 + s^2 = 1 ... (but this norm condition as pretty irrelevant due to freedom to rescale ... ??? ...) ?? we think that that's going to give ... ?? ....

?? 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 ...... ????

?? exponentials of (i-j,0) ... (r+s(i-j),0) .... ?? dual (r-s(i-j),0) .... ??? ....

(r+s(i-j),0)*(i,1)*(r-s(i-j),0) = ??

= (ri - s + sk,r + si - sj)*(r - si + sj,0) = (rri - rs + rsk + rs + ssi - ssj + rsk - ssj - ssi,rr + rsi - rsj + rsi - ss + ssk - rsj - ssk - ss) = (rri - 2ssj + 2rsk,rr - 2ss + 2rsi - 2rsj)

???? r,s = 1,1 .... ??? .... (i-2j+2k,-1+2i-2j) .... ??? ... ?? doesn't seem easy to visualize offhand ??? .....

?? "maximal associative subalgebra" .... ???? .... ?? includes unsplit quaternions though ??? ....

????? ... "modifying jordan product by declaring new neutral element" ... ??? "jordan triple system" ... ??? ... bounded symmetric domain .... ??? .....

??? "homming of qf spaces" .... ??? siegel .... ?? ... bit about "discriminant of binary quadratic form as almost sort of universal such" .... ??? ..... ?? hecke operator .... ??? .... ??? ....