I think Axiom/FriCAS thought about this problem hard and mostly solved it. The dominating CAS paradigm for most popular CASes is to assume everything is a real

I think Axiom/FriCAS thought about this problem hard and mostly solved it. The dominating CAS paradigm for most popular CASes is to assume everything is a real number and see how far you get. Later many major CASes bolted on features to say “well, except, this might be a complex number” and the like. This of course flies right in the face of the kind of thing you want to do. Axiom didn’t take that approach and instead created a robust system for specifying and implementing mathematical objects. I tried doing quarternion algebra in Maxima many moons ago and it was painful. They’ve since added packages for doing Clifford algebra but it’s not exactly well integrated in the rest of the machinery.