This can be better expressed in the geometric algebra by just considering a geometric algebra with a degenerate metric (ie. you use basis elements x̂, ŷ, ẑ and

This can be better expressed in the geometric algebra by just considering a geometric algebra with a degenerate metric (ie. you use basis elements x̂, ŷ, ẑ and ϵ̂ where x̂² = ŷ² = ẑ² = 1 and ϵ̂² = 0). Including factors of ϵ̂ in your rotors now is equivalent to a dual quaternion. This way, one gets the wonderful properties of geometric algebra and the useful properties of dual quaternions together.