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. [ABSTRACT FROM AUTHOR]