Subtle characteristic classes for Spin-torsors

Extending (Smirnov and Vishik, Subtle Characteristic Classes, arXiv:1401.6661), we obtain a complete description of the motivic cohomology with Z/2-coefficients of the Nisnevich classifying space of the spin group Spinn associated to the standard split quadratic form. This provides us with very simple relations among subtle Stiefel–Whitney classes in the motivic cohomology of Čech simplicial schemes associated to quadratic forms from I3, which are closely related to Spinn-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 Spin7 and G2, we describe completely the motivic cohomology ring of the Nisnevich classifying space of G2. The result in topology was obtained by Quillen (Math Ann 194:197–212, 1971).