Alright, make this a separate answer... there are tables of positive forms available up to dimension 5. So I revised some C++ programs and found the "odd" lattices that satisfy Allcock's rule, the ambient vector space is covered when one places an open ball of radius 1 around each lattice point of odd norm, then an open ball of radius $\sqrt 2$ around each lattice point of even norm (such as the origin). A lattice is called "borderline" if those open balls do not suffice but closed balls of the same radii do the job.
So I am posting the table, so far (dimension 1,2,3,4,5), at this LINK. One can see that these are integral lattices as the matrix entries listed (one matrix per line) gives inner products always integral, meaning all f_ij are even when i,j are distinct, but the lattices are "odd" in that at least one diagonal entry f_ii is odd.