?? trying to give nice axiomatic (...) characterization of cayley-dickson process as endofunctor of _alg with involution_ riding endofunctor "_ # k^2" ... equipped with embedding form identity functor .... ??? .... ??? .... ??? ....

?? confusion here about acting (that is, the z/2 action given by the involution ...) vs grading .... ??? ....

?? ok, probably some of the confusion was from some people saying "algebra with involution" for "algebra with anti-involution" ... which might arguably be defensible, though also arguably indefensible, since true "algebras with involution" are probably quite important in their own way ...

?? anyway, does this explain why for a moment i was expecting the product of imaginaries to be real, despite being somewhat familiar with "ij = k" ... ??? ....

?? but ... in the anti-involution case is there anything vaguely like a substitute for the product of imaginaries being real ??? .....

(xy)* = x* y* vs (xy)* = y* x* .... ??? ....

x imaginary <=> x* = -x ... ?? ...

(xy)* = (-x)(-y) = xy .... ???? ....

??? commutators and anticommutators here ??? ..... ??? anticommutators of imaginaries as real, commutators of imaginaries as imaginary ????? .... ???? ..... ??? did we in fact bump into "anticommutators of imaginaries as real" the other day, without quite completely noticing it ??? .... hmmmm ..... ????? ..... ?? in connection with .... ???? jordan product of traceless operators ..... ???? ..... ???? ..... ????? ..... ????? traceless vs symmetric ... ?? again ?? ... ??? .... ??? ....

???? commutator of real and real .... ??? or of real and imaginary .... ????? ......

x,y imaginary wrt anti-involution * .....

(xy+yx)* = (-y)(-x) + (-x)(-y) .... ??? so anticommutator of imaginaries is real ???

(xy-yx)* = (-y)(-x) - (-x)(-y) .... ??? so commutator of imaginaries is imaginary ???

x,y real wrt anti-involution * ....

(xy-yx)* = yx-xy ... ???? so commutator of reals really is imaginary here ???? ..... ???? so an associative alg with anti-involution gives a graded jordan algebra and a _shiftedly_ graded lie algebra ??? .... ..... ????? ..... and non-associative case ..... ?????? .....

?? relationship to other grading shifts ????? ..... ????? .....

??? generalizations to algebra with twisted (?? ...) order n endomorphism ??? ....

?? generalizations to algebras of arbitrary vsp-enriched operad (?? ...) ?? ... ??? ...

?? anyway, back to acting vs grading .... ??? .... ?? duality between them as somewhat broken ??? .... ?? in maybe interesting way ... ??? vague feeling of relationship to recent observations about ... ?? very different galois reps associated to (?? "fourier" ??? ...) dual bistable hopf algs ... ???? ..... ?? anyway, grading as always special case of acting via duality, but not generally vice versa .... ???? ......

?? nevertheless, might be interesting to look at grading version of cayley-dickson process .... ???? ..... ??? ..... ?????? ....... ....

?? conversely (??? ...), might be interesting to look at acting version of "superalgebra", trying to squeeze out some extra generality .... ???? .....