That's hardly the most straightforward or intuitive decomposition. In particular, it makes simple monoplanar rotations overly complicated, in exchange for makin

That's hardly the most straightforward or intuitive decomposition. In particular, it makes simple monoplanar rotations overly complicated, in exchange for making isoclinic rotations super simple--while rotors handle monoplanar rotations entirely naturally, with an obvious extension to isoclinic rotations. And monoplanar rotations are a much nicer primitive to work with.