That makes complete sense to me. But then later on they say "The output is a Geometric with a scalar component s and a bivector component ⇒c, which has 1 + 5 = 6 degrees of freedom so this system is not lossy! It should permit an unambiguous inversion operation!" If a bivector only has 3 degrees of freedom then the total is 4, which seems like it would be lossy?
I was also wondering this. But note that x^y is always perpendicular to x, so really only has two degrees of freedom while you need three to recover y (knowing x). Add in the dot product part to make up for it.
