I don’t know what was going through your head, but you really picked the wrong paradoxical decomposition, somehow you were able to make a new hamster but it wasn’t right at all, now Dini’s all into wearing leather and riding around on his hog with all his hamster friends. I don’t know, unless you’re comfortable with Dini hanging out with that crowd, I’d try another decomposition!
A1=S(a)M, A2=S(b^−1)MUM\B , A3=S(a^−1)M, A4=S(b)BUMUB
A1=S(b^−1)M\B, A2=S(a)MUMUB, A3=S(b)B, A4=S(a^−1)M