We shall prove that the map is bijective for and surjective for . Here is an arbitrary Pythagorean field and the special orthogonal group is the subgroup of -linear automorphisms over with determinant one which preserve the Euclidean quadratic form . It is derived from the homological stability of the orthogonal groups with twisted coefficients .
"On homological stability for orthogonal and special orthogonal groups." Kyoto J. Math. 55 (3) 627 - 640, September 2015. https://doi.org/10.1215/21562261-3089109