Azumaya Algebras With Orthogonal Involution Admitting an Improper Isometry

Let $(A,\sigma )$ be an Azumaya algebra with orthogonal involution over a ring $R$ with $2\in R^{\times }$. We show that if $(A,\sigma )$ admits an improper isometry, i.e., an element $a\in A$ with $\sigma (a)a=1$ and ${\mathrm{N}\mathrm{r}\mathrm{d}}_{A\mathrm{/}R}(a)=-1$, then the Brauer class of $A$ is trivial. An analogue of this statement also holds for Azumaya algebras with quadratic pair when $2\mathrm{\notin }{R}^{\times }$. We also show that at this level of generality, the hypotheses do not guarantee that $A$ is a matrix algebra over $R$.