A curious example on classifying stacks

In Jarod Alper's note i found the following exercise: Consider two nondegenerate quadratic forms q, q,' of the same rank over a field. Then their classifying stacks BO(q) and BO(q') are isomorphic. This seemed really counterintuitive. We look at quotients of the same space by two non-isomorphic groups and we get the same space.

So here is my attempt at proving this fact. Maybe this is useful to someone.

classifying_stack.pdf

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