Roughly speaking, the weak local global principle in algebraic K-theory assures that two locally isomorphic modules or forms differ- -on the level of Grothendieck rings - - by a nilpotent element only. This paper contains a thorough investigation of this rather useful principle in various cases, including Witt and Witt-Grothendieck rings and their equivariant generalizations. The proofs are based on an abstract categorical "localization lemma" which reduce s the proof in each special case to the verification of two simple assertions. In the .case of Witt rings the results are applied to deduce Pfister's basic structure theorems for Witt rings over arbitrary commutative rings from the corresponding statements for Witt rings over local rings. In the last section explicit isomorphisms of certain modules and forms, derived from locally isomorphic modules or forms in a canonical way, are exhibited. There are further possible applications of the results of this paper towards the theory of signatures of Witt rings and towards integral representation theory.