  We prove that Toen's secondary Grothendieck ring is isomorphic to the
Grothendieck ring of smooth proper pretriangulated dg categories previously
introduced by Bondal, Larsen and Lunts. Along the way, we show that those short
exact sequences of dg categories in which the first term is smooth proper and
the second term is proper are necessarily split. As an application, we prove
that the canonical map from the derived Brauer group to the secondary
Grothendieck ring has the following injective properties: in the case of a
commutative ring of characteristic zero, it distinguishes between dg Azumaya
algebras associated to non-torsion cohomology classes and dg Azumaya algebras
associated to torsion cohomology classes (=ordinary Azumaya algebras); in the
case of a field of characteristic zero, it is injective; in the case of a field
of positive characteristic p>0, it restricts to an injective map on the
p-primary component of the Brauer group.
