A new formulation of what may be called the “fundamental theorem of the theory of relativity” is presented and proved in -space-time, based on the full classification of special transformations and the corresponding velocity addition laws. A system of axioms is introduced and discussed leading to the result, and a study is made of several variants of that system. In particular the status of the group axiom is investigated with respect to the condition of the two-way isotropy of light. Several (...) issues which are ignored or misunderstood in the literature are emphasized. (shrink)