The first term is conserved as expected. The vector portion of the square is frame dependent. L is too complex for simple calculations. Mathematica does the grunge work. In these cases, the boost quaternion L becomes a very simple.
Redshifts have a plus instead of the minus.
Over 50 problems in a sophomore-level relativistic mechanics class have been solved using quaternions. Problems in special relativity can be solved either using 4-vectors, the Minkowski metric and the Lorentz group, or using quaternions. No experimental difference between the two methods has been presented. At this point the difference is in the mathematical foundations. An immense amount of work has gone into the study of metrics, particular in the field of general relativity.