Most Niemeier matrices are characterized by their number of "roots"
(vectors of norm 2).  In this case we're lucky: we find
(e.g. using the **qfminim** function in **gp**) that there are 
$528$ roots, and thus that each of the simple root lattices
contained in the lattice has Coxeter number $528/24 = 22$;
and this determines the lattice uniquely: it is the one with
root lattice $D_{12}^2$.

In some cases there are two Niemeier lattices with the same
Coxeter number, but they can be distinguished by the index of the 
sublattice generated by the roots (which can be obtained in **gp**
with the command **matdet(matrixqz(qfminim(G)[3],-1))**
assuming **G** is the name of the Gram matrix).

To partly answer your earlier question:
The Nebe-Sloane "Catalogue of Lattices" gives rational bases for all the
Niemeier lattices (start from the second item in the
<a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/index.html#24D">$
D=24$ page</a>),
though the coordinates are given as floating-point real numbers
so they might not be as useful as you might like.