Subtle characteristic classes for $Spin$-torsors

Extending [14], we obtain a complete description of the motivic cohomology with $\mathbb{Z}\mathrm{/}2$-coefficients of the Nisnevich classifying space of the spin group $Spi{n}_n$ associated to the standard split quadratic form. This provides us with very simple relations among subtle Stiefel-Whitney classes in the motivic cohomology of \v{C}ech simplicial schemes associated to quadratic forms from $I^3$, which are closely related to $Spi{n}_n$-torsors over the point. These relations come from the action of the motivic Steenrod algebra on the second subtle Stiefel-Whitney class. Moreover, exploiting the relation between $Spi{n}_7$ and $G_2$, we describe completely the motivic cohomology ring of the Nisnevich classifying space of $G_2$. The result in topology was obtained by Quillen in [13].