You seem to be completely missing the point. There is no complement being used here whatsoever. The map we are after, which we use solely to define the regressive product, has only one (geometric, not algebraic) requirement :
- it must map ‘points on a line’ to ‘lines through a point’ (in 2D)
The J map is the simpelest, trivial map that does that. It has no sign swaps anywhere and it uses no complements. It is an involution and is the most efficient way to define the regressive product.
Basis order and metric are completely irrelevant - you can add factors as much as you like, you’d only be correct if you take them back out in the very next instruction after the outer product.
(since we apply J, do one outer product and apply the inverse of J (which is J again, its an involution)) to define the regressive product:
a \vee b = J^{-1}(J(a) \wedge J(b))
Unrelated to the definition of the regressive product, you say
As I explained many times, the best and most straight forward definition is the following: \star\omega = \tilde\omega I, which is simply the reverse element times the pseudoscalar with the geometric product.
This clearly is wrong when there is a degenerate metric. i.e. that map clearly is not an anti-involution. For proof calculate, in \mathbb R_{2,0,1}, the double application of your map on e_{01} where e_0^2 = 0 and e_1^2 = 1 :
“simply the reverse times the pseudoscalar” : \tilde e_{01} e_{012} = e_{10} e_{012} = 0
This is why people keep ‘ignoring’ your definition in PGA. (Hope that helps cause I’m tired of hearing how many times you’ve explained it - when I’ve never seen you explain it).