The “Transformations pdf” file embedded below presents a derivation for the coordinate transformations between two inertial reference frames in special relativity theory, where the second frame is seen from the first frame as moving with constant velocity ( and also the first from the second) under the assumption that if a particle is moving uniformly rectilineal as it is seen in the one frame then it will be seen moving uniformly rectilineal also in the other one frame. We assume also the continuity of coordinate transformations between inertial frames and that for any event in space-time we can consider along any spatial direction the event separated from it by a light signal, such that light signals travel uniformly rectilineal with a velocity that has the same modulus in all inertial reference frames. Further we suppose that the causality principle is valid, so that if an event precedes an other event at the same spatial point as it is seen in one inertial reference frame, then that one event precedes the other one event as it is seen in any other inertial reference frame.
At the end, the definition of the restricted Lorentz group and of the Poincare group follows.
This chapter you can also find in the post
Notes on … quantum field theory