  We consider Borcherds superalgebras obtained from semisimple
finite-dimensional Lie algebras by adding an odd null root to the simple roots.
The additional Serre relations can be expressed in a covariant way. The
spectrum of generators at positive levels are associated to partition functions
for a certain set of constrained bosonic variables, the constraints on which
are complementary to the Serre relations in the symmetric product. We give some
examples, focusing on superalgebras related to pure spinors, exceptional
geometry and tensor hierarchies, of how construction of the content of the
algebra at arbitrary levels is simplified.
