Additivity for the Motivic Trace and the Motivic Euler Characteristic

In this paper, we settle an open conjecture regarding the assertion that the Euler-characteristic of $\mathrm{G}\mathrm{/}\text{NT}$ for a split reductive group scheme $\mathrm{G}$ and the normalizer of a split maximal torus $\text{NT}$ 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.