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

Principal subspaces of basic modules for twisted affine Lie algebras, $q$-series multisums, and Nandi's identities