  In this paper, we settle an open conjecture regarding the assertion that the
Euler-characteristic of $\rmG/\NT$ for a split reductive group scheme $\rmG$
and the normalizer of a split maximal torus $\NT$ over a field is $1$ in the
Grothendieck-Witt ring with the characteristic exponent of the field inverted,
under the assumption that the base field contains a $\sqrt -1$. Numerous
applications of this to splittings in the motivic stable homotopy category and
to Algebraic K-Theory are worked out in several related papers by Gunnar
Carlsson and the authors.
