Well yeah, it is. The point I wanted to make is that these isomorphism are "exceptional"[1] and only hold for the lower dimensional groups. The general Spin gro

Well yeah, it is. The point I wanted to make is that these isomorphism are "exceptional"[1] and only hold for the lower dimensional groups. The general Spin group and quaterions are very different objects. [1]: https://en.wikipedia.org/wiki/Exceptional_isomorphism https://en.wikipedia.org/wiki/Exceptional_isomorphism