  We study the algebraic Brauer classes on open del Pezzo surfaces of degree
$4$. I.e., on the complements of geometrically irreducible hyperplane sections
of del Pezzo surfaces of degree $4$. We show that the $2$-torsion part is
generated by classes of two different types. Moreover, there are two types of
$4$-torsion classes. For each type, we discuss methods for the evaluation of
such a class at a rational point over a $p$-adic field.
