Look at the nonrelativistic limit. In the nonrelativistic limit, the lab kinetic energy equals the kinetic energy within the center of mass frame plus the kinetic energy of all the individual masses in the center of mass frame. There are a few tools required to solve problems in special relativity using quaternions to characterize events in spacetime. The most basic are a round value for c and gamma. Define a function for quaternions using its matrix representation.
Redshifts have a plus instead of the minus.