Vertex-algebraic structure of the principal subspaces of certain image-modules, II: Higher-level case

We give an a priori proof of the known presentations of (that is, completeness of families of relations for) the principal subspaces of all the standard image-modules. These presentations had been used by Capparelli, Lepowsky and Milas for the purpose of obtaining the classical Rogers–Selberg recursions for the graded dimensions of the principal subspaces. This paper generalizes our previous paper.