I'm currently reading Conway and Sloane's Sphere Packings, Lattices and Groups. It is a great read for facts, but they only sketch proofs at a high level.
Within it, they define the lattice $D_n$ to be the set of all points with even coordinates. Another similar collection of points is $$D_n^+ = D_n \cup \left\{D_n+\vec{\frac{1}{2}}\right\}$$ One can show that this is a lattice only in even dimensions by noticing that $-\vec{1/2}$ can only be constructed in even dimensions. It is then stated without proof that this lattice is congruence to $\mathbb Z^4$ in 4 dimensions. I was wondering if anybody could provide a proof of this amazing fact? What about for other dimensions? I have found a basis with determinant 1, but the coefficients are rational, not integral.