We introduce what we call the principal subalgebra of a lattice vertex (super) algebra associated to an arbitrary Z-basis of the lattice. In the first part (to appear), the second author considered the case of positive bases and found a description of the principal subalgebra in terms of generators and relations. Here, in the most general case, we obtain a combinatorial basis of the principal subalgebra W_L and of related modules. In particular, we substantially generalize several results in Georgiev, 1996, covering the case of the root lattice of type A_n, as well as some results from Calinescu, Lepowsky and Milas, 2010. We also discuss principal subalgebras inside certain extensions of affine W-algebras coming from multiples of the root lattice of type A_n.

The first author graciously acknowledges support from NSA and NSF grants.