On the Motivic Class of the Classifying Stack of $G_2$ and the Spin Groups

We compute the class of the classifying stack of the exceptional algebraic group G2 and of the spin groups Spin7 and Spin8 in the Grothendieck ring of stacks, and show that they are equal to the inverse of the class of the corresponding group. Furthermore, we show that the computation of the motivic classes of the stacks ℬSpinn can be reduced to the computation of the classes of ℬΔn⁠, where Δn⊂Pinn is the “extraspecial 2-group”, the preimage of the diagonal matrices under the projection Pinn→On to the orthogonal group.