The definitions of quaternions and dual quaternions look particularly neat. Are there other (non-"geometric algebra") ways to define (dual) quaternions without

The definitions of quaternions and dual quaternions look particularly neat. Are there other (non-"geometric algebra") ways to define (dual) quaternions without reference to a basis? Edit: I guess the "scalar and vector parts" definition at Wikipedia[1] doesn't use a basis, but it's not exactly pretty... [1]: https://en.wikipedia.org/wiki/Quaternion#Scalar_and_vector_parts https://en.wikipedia.org/wiki/Quaternion#Scalar_and_vector_p...