No, I was referring to the group of non-zero quaternions. There's a double-cover from the group U of unit quaternions to SO(3) though. If you care about SE(3),

No, I was referring to the group of non-zero quaternions. There's a double-cover from the group U of unit quaternions to SO(3) though. If you care about SE(3), there's an obvious surjection U x R^3 -> SE(3) where the domain is "a translation following a rotation". This is a homomorphism of Lie groups. I just don't get what you were going for with "use Lie groups instead". Everything here appears to already be a Lie group. How am I supposed to use that?