Abstract
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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