Our purpose in this work is to extend and unify the intrinsic axiomatic perspective on spectral decompositions. In such extension we wish to consider the widest feasible generalization of the notion "spectral decomposition" in order to learn more about the special cases. In this spirit we start in Chapter II the study of the most abstract form of spectral decomposition so that when we come to the more special theory of "decomposable operators" (Chapter IV) we find that many of the known results of the latter theory are easy consequences of the preceding material. More importantly, however, we obtain solutions to deep problems which have been open and vigorously studied (e.g. the dual theory).