On the motivic class of an algebraic group

Abstract

Let $F$ be a field of characteristic zero admitting a biquadratic field extension. We give an example of a torus $G$ over $F$ whose classifying stack $BG$ is stably rational and such that $\left\{BG\right\}\ne {\left\{G\right\}}^{-1}$ in the Grothendieck ring of algebraic stacks over $F$. We also give an example of a finite étale group scheme $A$ over $F$ such that $BA$ is stably rational and $\left\{BA\right\}\ne 1$.