As part of a program for constructing invariants of enumerative problems with values in the Grothendieck-Witt ring, refining the classical integer invariants, we are developing a calculus of characteristic classes of vector bundles with values in the cohomology of the sheaf of Witt rings. This cohomology theory is the ``real twin’’ of the classical theory of the Chow ring and the more modern motivic cohomology, and the characteristic classes are algebro-geometric versions of the theory of Pontryagin classes and Euler classes. Building on Ananyevskiy’s SL_2-splitting principle, we will describe a splitting principle that enables one to have a calculus of these characteristic classes, where the normalizer of the usual torus in SL_2 replaces G_m,.