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We compute the class of the classifying stack of the exceptional algebraicgroup $G_2$ and of the spin groups $\mathrm{Spin}_7$ and $\mathrm{Spin}_8$ inthe Grothendieck ring of stacks, and show that they are equal to the inverse ofthe class of the corresponding group. Furthermore, we show that the computationof the motivic classes of the stacks $\mathscr{B}\mathrm{Spin}_n$ can bereduced to the computation of the classes of $\mathscr{B} \Delta_n$, where$\Delta_n\subset \mathrm{Pin}_n$ is the "extraspecial $2$-group", the preimageof the diagonal matrices under the projection $\mathrm{Pin}_n\to \mathrm{O}_n$to the orthogonal group.

On the motivic class of the classifying stack of $G_2$ and the spin groups