The determinant on 2 by 2 matrices of trace zero gives a quadratic form. Linear algebra shows that its orthogonal group is the matrix algebra automorphisms together with scalars of square 1. As this holds over all commutative rings, descent theory can be applied to it to get a quick classification of all nonsingular quadratic forms of rank 3. Further descent arguments show how this classification yields the Witt invariant and discriminant. © 1990.
Waterhouse, W. C. (1990). Classifying nonsingular quadratic forms of rank three. Linear Algebra and Its Applications, 142(C), 55–61. https://doi.org/10.1016/0024-3795(90)90255-B