On the non-neutral component of outer forms of the orthogonal group

Let $(A,σ)$ be a central simple algebra with an orthogonal involution. It is well-known that $O(A,σ)$ contains elements of reduced norm $-1$ if and only if the Brauer class of $A$ is trivial. We generalize this statement to Azumaya algebras with orthogonal involution over semilocal rings, and show that the "if" part fails if one allows the base ring to be arbitrary.

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