On the motivic class of an algebraic group

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 $\{BG\}\{G\}\neq 1$ in the Grothendieck ring of algebraic stacks over $F$. We also give an example of a finite \'etale group scheme $A$ over $F$ such that $BA$ is stably rational and $\{BA\}\neq 1$.