I do not see the point.
When you study complex number you do not write the product as
z_1*z_2 = ( ac - bd , ad + bc )
with z_1 = ( a , b ) and z_2 = (c , d )
to maintain your brain, all the life, in pairs of real numbers.
You just write
a+bi and c+di to get
ac - bd + i ( ad + bc )
At this level you are combining a "real number" plus an "imaginary number".
This mean two "different things".
Thus
AB := A.B + A/\B
are another pair of different things.
Certainly, you can start with the geometric product to separate in a
symmetric plus
antisymmetric part. But it is just another approach.
Thanks
Arnulfo Castellanos-Moreno
This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.