TITLE: Even unimodular lattices with root system $32 A_1$
QUESTION [5 upvotes]: I'm studying Venkov's proof of the classification of even unimodular rank 24 lattices, and it prompted the following question.
For an even unimodular lattice $L$, let $R(L)= \{ x \in L : (x,x) =2\}$ be its root system. 
Question. Can one "roughly" classify, up to isomorphism, all rank $32$ even unimodular lattices with root system $32 A_1$? Or are there too many of these? (Note: there are at least $10^8$ even unimodular lattices of rank 32.) 
Remark. There are no even unimodular lattices of rank 8 and 16 with root system $8 A_1$ and $16A_1$, respectively. There is a unique rank $24$ even unimodular lattice with root system $24 A_1$ by the work of Niemeier (and Venkov).
Any comments are appreciated!

REPLY [2 votes]: Another construction (basically, different terminology, but the same outcome) is the maximal (of rank 16) isotropic subgroups in the discriminant group (in the sense of Nikulin) of $32A_1$, which is $32\langle\frac12\rangle$, not containing a sum of four generators. This is a finite problem, but I wouldn't do that manually.