I noticed the following problem in my code:
 There are 2 transformation matrices $ A, B in mathbb {R} ^ {4 times 4} $

$ A, B $ have this shape $ begin {bmatrix} R cdot s && t \ 0 && 1 end {bmatrix} $

The goal is to decompose $ A ^ { 1} cdot B $ in the translation $ t in mathbb {R} ^ 3 $, Rotation $ q in text {quaternion} $ and scale $ s in mathbb {R} ^ 3 $
For the safety check I introduce Round trip (Decomposition and compilation of a matrix):
 $ comp (dec (A)) == A $
 $ comp (dec (B)) == B $
 $ comp (dec (A cdot B)) == A cdot B $
However, if I try to make the tour with
 $ comp (dec (A ^ { 1} cdot B)) $
Then I get a relatively large error in the rotation part. What is the reason for this phenomenon? Is it numerically unstable or is the inversion too inaccurate?
Many Thanks
Note: The decomposition is done as described here