where (t, x, y, z) are the spacetime coordinates used to define events in one frame, and (t ′, x ′, y ′, z ′) are the coordinates in another frame. First one observes that (D2) is satisfied if an arbitrary 4 -tuple b of numbers are added to events a 1 and a 2. Such transformations are called spacetime translations and are not dealt with further here. Then one observes that a linear solution preserving the origin of the simpler problem solves the general problem too: