A 1926 theorem of I. Schur concerns partitions of an integer \(n\) into parts congruent to \(\pm 1\pmod 6\). This leads to the consideration of the infinite product \[ \prod_{n=1}^{\infty} \frac{1}{(1-q^{6n-1})(1-q^{6n-5})}\tag{1} \] Any nontrivial rewriting of such an infinite product is potentially important as it might yield a partition identity, or, in some cases, provide hints to the construction of a representation of an affine Lie algebra [see \textit{J. Lepowsky} and \textit{R. L. Wilson}, Invent. Math. 77, 199--290 (1984; Zbl 0577.17009)]. In the present work, the authors construct an explicit vertex operator representation of the affine Lie algebra \(C_3^{(1)}\) and some Lepowsky-Wilson \(Z\)-operators to prove a first rewriting of Schur's infinite product as a sum of two other infinite products. Next, the authors use \textit{V. Kac}'s and \textit{M. Wakimoto}'s character formula for an admissible module [Proc. Natl. Acad. Sci. USA 85, 4956--4960 (1988; Zbl 0652.17010)] for the affine Lie algebra \(A_1^{(1)}\) and the quintuple product identity to obtain a different rewriting of the same infinite product (1). Finally, by using the Rogers-Ramanujan partition identities they obtain a combinatorial interpretation of these results.