  Generalizing a theorem of Albert, Saltman showed that an Azumaya algebra $A$
over a ring represents a $2$-torsion class in the Brauer group if and only if
there is an algebra $A'$ in the Brauer class of $A$ admitting an involution of
the first kind. Knus, Parimala, and Srinivas later showed that one can choose
$A'$ such that $\mathrm{deg}\, A'=2\mathrm{deg}\, A$. We show that
$2\mathrm{deg}\, A$ is the lowest degree one can expect in general.
Specifically, we construct an Azumaya algebra $A$ of degree $4$ and period $2$
such that the degree of any algebra $A'$ in the Brauer class of $A$ admitting
an involution is divisible by $8$.
  Separately, we provide examples of split and non-split Azumaya algebras of
degree $2$ admitting symplectic involutions, but no orthogonal involutions.
These stand in contrast to the case of central simple algebras of even degree
over fields, where the presence of a symplectic involution implies the
existence of an orthogonal involution and vice versa.
