Perhaps I could factor orthogonal matrices into products for each of which I knew the Clifford number. Perhaps I could do the opposite but that seems harder, especially since Clifford algebras are not division algebras.
Factoring an orthogonal matrix into simple rotations is not unique. Would different factorizations produce different Clifford numbers, beyond the sign ambiguity? That sounds like a feasible programming project!
Given an orthogonal matrix (OM) we multiply it, on left and right, by a sequence of standard OMs to put it into a form where it is a matrix of zeros except for 2 by 2 matrices down the diagonal of the form:
cosθ_{1}  sinθ_{1}  0  0  .... 
−sinθ_{1}  cosθ_{1}  0  0  ... 
0  0  cosθ_{2}  sinθ_{2}  ... 
0  0  −sinθ_{2}  cosθ_{2}  ... 
...  ...  ...  ...  ... 
(M_{mn}(θ))_{ij} = 
