Suppose we have a full rank, integer sublattice $L$ of the integer lattice $\mathbb Z^d$, where we fix the dimension $d$. Consider the Gram matrix $M$ of $L$, relative to some basis for $L$, and reduce all the entries of $M$ mod $4$. Is there a nice clean description of all the finite types of such mod $4$ reductions of Gram matrices, as we vary over all full rank integer sublattices $L \subset \mathbb Z^d$ while keeping the dimension $d$ fixed?
There is a theorem that attempts to describe the Gram matrix of an integer lattice mod powers of $2$, in J.W.S. Cassells' book ``Rational quadratic forms", Section VIII.4, p. 117 in this book. But in the beginning of that section he writes "...This section is only for the masochistic". I would be very grateful if anyone has found a cleaner description and/or proof, at least in the mod $4$ case.