The claim that geometric algebra is not rigorous made me confused. But the lack of proper references is much more confusing to me. Back in the days when I was doing research in Chevalley groups the definitive and most cited source was (and still is) Geometric algebra by E. Artin. And there are some more recent treatments recognized in math community, e.g. classical groups and geometric algebra by L. Groove and the geometry of the classical groups by D. Taylor. At least this books made everything rigorous and clear. But may be there is some other field with the same title?

I've heard about book by Doran and always thought that it's more about applications.