  We provide an observation relating several known and conjectured $q$-series
identities to the theory of principal subspaces of basic modules for twisted
affine Lie algebras. We also state and prove two new families of $q$-series
identities. The first family provides quadruple sum representations for Nandi's
identities, including a manifestly positive representation for the first
identity. The second is a family of new mod 10 identities connected with
principal characters of level 4 integrable, highest-weight modules of
$\mathrm{D}_4^{(3)}$.
