E1, E2, E3 = basis vectors, geometric multiplication rules:
given u and v are basis vectors,
uu = 1
uv = -vu

want to compute: a b (rotor composition) and a b a* (rotor rotation)
where a, b are rotors, a* = reverse(a)

a = Sa + BaxyE1E2 + BaxzE1E3 + BayzE2E3
b = Sb + BbxyE1E2 + BbxzE1E3 + BbyzE2E3
a* = Sa - BaxyE1E2 - BaxzE1E3 - BayzE2E3

-------------

a b = (Sa + BaxyE1E2 + BaxzE1E3 + BayzE2E3)(Sb + BbxyE1E2 + BbxzE1E3 + BbyzE2E3)

= SaSb + SaBbxyE1E2 + SaBbxzE1E3 + SaBbyzE2E3
+ SbBaxyE1E2 + BaxyBbxyE1E2E1E2 + BaxyBbxzE1E2E1E3 + BaxyBbyzE1E2E2E3
+ SbBaxzE1E3 + BaxzBbxyE1E3E1E2 + BaxzBbxzE1E3E1E3 + BaxzBbyzE1E3E2E3
+ SbBayzE2E3 + BayzBbxyE2E3E1E2 + BayzBbxzE2E3E1E3 + BayzBbyzE2E3E2E3

= SaSb + SaBbxyE1E2 + SaBbxzE1E3 + SaBbyzE2E3
+ SbBaxyE1E2 - BaxyBbxy - BaxyBbxzE2E3 + BaxyBbyzE1E3
+ SbBaxzE1E3 + BaxzBbxyE2E3 - BaxzBbxz - BaxzBbyzE1E2
+ SbBayzE2E3 - BayzBbxyE1E3 + BayzBbxzE1E2 - BayzBbyz

= SaSb - BaxyBbxy - BaxzBbxz - BayzBbyz
+ SaBbxyE1E2 + SbBaxyE1E2 - BaxzBbyzE1E2 + BayzBbxzE1E2
+ SaBbxzE1E3 + SbBaxzE1E3 + BaxyBbyzE1E3 - BayzBbxyE1E3
+ SaBbyzE2E3 + SbBayzE2E3 - BaxyBbxzE2E3 + BaxzBbxyE2E3

-------------

a b a* = (a b)(Sa - BaxyE1E2 - BaxzE1E3 - BayzE2E3)

= (SaSb - BaxyBbxy - BaxzBbxz - BayzBbyz
+ SaBbxyE1E2 + SbBaxyE1E2 - BaxzBbyzE1E2 + BayzBbxzE1E2
+ SaBbxzE1E3 + SbBaxzE1E3 + BaxyBbyzE1E3 - BayzBbxyE1E3
+ SaBbyzE2E3 + SbBayzE2E3 - BaxyBbxzE2E3 + BaxzBbxyE2E3)
* (Sa - BaxyE1E2 - BaxzE1E3 - BayzE2E3)


= SaSaSb - SaBaxyBbxy - SaBaxzBbxz - SaBayzBbyz
+ SaSaBbxyE1E2 + SaSbBaxyE1E2 - SaBaxzBbyzE1E2 + SaBayzBbxzE1E2
+ SaSaBbxzE1E3 + SaSbBaxzE1E3 + SaBaxyBbyzE1E3 - SaBayzBbxyE1E3
+ SaSaBbyzE2E3 + SaSbBayzE2E3 - SaBaxyBbxzE2E3 + SaBaxzBbxyE2E3

- SaSbBaxyE1E2 + BaxyBaxyBbxyE1E2 + BaxyBaxzBbxzE1E2 + BaxyBayzBbyzE1E2
- SaBaxyBbxyE1E2E1E2 - SbBaxyBaxyE1E2E1E2 + BaxyBaxzBbyzE1E2E1E2 - BaxyBayzBbxzE1E2E1E2
- SaBaxyBbxzE1E3E1E2 - SbBaxyBaxzE1E3E1E2 - BaxyBaxyBbyzE1E3E1E2 + BaxyBayzBbxyE1E3E1E2
- SaBaxyBbyzE2E3E1E2 - SbBaxyBayzE2E3E1E2 + BaxyBaxyBbxzE2E3E1E2 - BaxyBaxzBbxyE2E3E1E2

- SaSbBaxzE1E3 + BaxyBaxzBbxyE1E3 + BaxzBaxzBbxzE1E3 + BaxzBayzBbyzE1E3
- SaBaxzBbxyE1E2E1E3 - SbBaxyBaxzE1E2E1E3 + BaxzBaxzBbyzE1E2E1E3 - BaxzBayzBbxzE1E2E1E3
- SaBaxzBbxzE1E3E1E3 - SbBaxzBaxzE1E3E1E3 - BaxyBaxzBbyzE1E3E1E3 + BaxzBayzBbxyE1E3E1E3
- SaBaxzBbyzE2E3E1E3 - SbBaxzBayzE2E3E1E3 + BaxyBaxzBbxzE2E3E1E3 - BaxzBaxzBbxyE2E3E1E3

- SaSbBayzE2E3 + BaxyBayzBbxyE2E3 + BaxzBayzBbxzE2E3 + BayzBayzBbyzE2E3
- SaBayzBbxyE1E2E2E3 - SbBaxyBayzE1E2E2E3 + BaxzBayzBbyzE1E2E2E3 - BayzBayzBbxzE1E2E2E3
- SaBayzBbxzE1E3E2E3 - SbBaxzBayzE1E3E2E3 - BaxyBayzBbyzE1E3E2E3 + BayzBayzBbxyE1E3E2E3
- SaBayzBbyzE2E3E2E3 - SbBayzBayzE2E3E2E3 + BaxyBayzBbxzE2E3E2E3 - BaxzBayzBbxyE2E3E2E3


= SaSaSb - SaBaxyBbxy - SaBaxzBbxz - SaBayzBbyz
+ SaSaBbxyE1E2 + SaSbBaxyE1E2 - SaBaxzBbyzE1E2 + SaBayzBbxzE1E2
+ SaSaBbxzE1E3 + SaSbBaxzE1E3 + SaBaxyBbyzE1E3 - SaBayzBbxyE1E3
+ SaSaBbyzE2E3 + SaSbBayzE2E3 - SaBaxyBbxzE2E3 + SaBaxzBbxyE2E3

- SaSbBaxyE1E2 + BaxyBaxyBbxyE1E2 + BaxyBaxzBbxzE1E2 + BaxyBayzBbyzE1E2
+ SaBaxyBbxy + SbBaxyBaxy - BaxyBaxzBbyz + BaxyBayzBbxz
- SaBaxyBbxzE2E3 - SbBaxyBaxzE2E3 - BaxyBaxyBbyzE2E3 + BaxyBayzBbxyE2E3
+ SaBaxyBbyzE1E3 + SbBaxyBayzE1E3 - BaxyBaxyBbxzE1E3 + BaxyBaxzBbxyE1E3

- SaSbBaxzE1E3 + BaxyBaxzBbxyE1E3 + BaxzBaxzBbxzE1E3 + BaxzBayzBbyzE1E3
+ SaBaxzBbxyE2E3 + SbBaxyBaxzE2E3 - BaxzBaxzBbyzE2E3 + BaxzBayzBbxzE2E3
+ SaBaxzBbxz + SbBaxzBaxz + BaxyBaxzBbyz - BaxzBayzBbxy
- SaBaxzBbyzE1E2 - SbBaxzBayzE1E2 + BaxyBaxzBbxzE1E2 - BaxzBaxzBbxyE1E2

- SaSbBayzE2E3 + BaxyBayzBbxyE2E3 + BaxzBayzBbxzE2E3 + BayzBayzBbyzE2E3
- SaBayzBbxyE1E3 - SbBaxyBayzE1E3 + BaxzBayzBbyzE1E3 - BayzBayzBbxzE1E3
+ SaBayzBbxzE1E2 + SbBaxzBayzE1E2 + BaxyBayzBbyzE1E2 - BayzBayzBbxyE1E2
+ SaBayzBbyz + SbBayzBayz - BaxyBayzBbxz + BaxzBayzBbxy


= SaSaSb - SaBaxyBbxy - SaBaxzBbxz - SaBayzBbyz
+ SaBaxyBbxy + SbBaxyBaxy - BaxyBaxzBbyz + BaxyBayzBbxz
+ SaBaxzBbxz + SbBaxzBaxz + BaxyBaxzBbyz - BaxzBayzBbxy
+ SaBayzBbyz + SbBayzBayz - BaxyBayzBbxz + BaxzBayzBbxy

+(SaSaBbxy + SaSbBaxy - SaBaxzBbyz + SaBayzBbxz
- SaSbBaxy + BaxyBaxyBbxy + BaxyBaxzBbxz + BaxyBayzBbyz
+ SaBayzBbxz + SbBaxzBayz + BaxyBayzBbyz - BayzBayzBbxy
- SaBaxzBbyz - SbBaxzBayz + BaxyBaxzBbxz - BaxzBaxzBbxy)E1E2

+(SaSaBbxz + SaSbBaxz + SaBaxyBbyz - SaBayzBbxy
- SaSbBaxz + BaxyBaxzBbxy + BaxzBaxzBbxz + BaxzBayzBbyz
+ SaBaxyBbyz + SbBaxyBayzE1E3 - BaxyBaxyBbxz + BaxyBaxzBbxy
- SaBayzBbxy - SbBaxyBayzE1E3 + BaxzBayzBbyz - BayzBayzBbxz)E1E3

+(SaSaBbyz + SaSbBayz - SaBaxyBbxz + SaBaxzBbxy
- SaSbBayz + BaxyBayzBbxy + BaxzBayzBbxz + BayzBayzBbyz
- SaBaxyBbxz - SbBaxyBaxz - BaxyBaxyBbyz + BaxyBayzBbxy
+ SaBaxzBbxy + SbBaxyBaxz - BaxzBaxzBbyz + BaxzBayzBbxz)E2E3


= SaSaSb 
+ SbBaxyBaxy 
+ SbBaxzBaxz
+ SbBayzBayz

+(SaSaBbxy + 2SaBayzBbxz
+ BaxyBaxyBbxy + 2BaxyBaxzBbxz + 2BaxyBayzBbyz
- BayzBayzBbxy
- 2SaBaxzBbyz - BaxzBaxzBbxy)E1E2

+(SaSaBbxz + 2SaBaxyBbyz - 2SaBayzBbxy
+ 2BaxyBaxzBbxy + BaxyBaxzBbxz + 2BaxzBayzBbyz
- BaxyBaxyBbxz - BayzBayzBbxz)E1E3

+(SaSaBbyz - 2SaBaxyBbxz + 2SaBaxzBbxy
+ 2BaxyBayzBbxy + 2BaxzBayzBbxz + BayzBayzBbyz
- BaxyBaxyBbyz
- BaxzBaxzBbyz)E2E3

= (Sa^2 + Baxy^2 + Baxz^2 + Bayz^2) * Sb

+((Sa^2 + Baxy^2 - Baxz^2 - Bayz^2) * Bbxy
+ (BaxyBaxz + SaBayz) * 2Bbxz
+ (BaxyBayz - SaBaxz) * 2Bbyz
)E1E2

+((Sa^2 - Baxy^2 + Baxz^2 - Bayz^2) * Bbxz
+ (BaxyBaxz - SaBayz) * 2Bbxy
+ (BaxzBayz + SaBaxy) * 2Bbyz
)E1E3

+((Sa^2 - Baxy^2 - Baxz^2 + Bayz^2) * Bbyz
+ (BaxyBayz + SaBaxz) * 2Bbxy
+ (BaxzBayz - SaBaxy) * 2Bbxz
)E2E3