Integral quadratic lattice ($L\cong {\mathbb{Z}}^n,B:L\times L\to \mathbb{Z}$)

An integral quadratic lattice is a finite-rank free abelian group with a nondegenerate integral symmetric bilinear form. Its norm convention here is $q(x)=B(x,x)$. A basis gives an integral Gram matrix; basis changes preserve its determinant up to multiplication by the square of a unit.

Every vector in an even lattice has even integral norm. The E8 lattice and the hyperbolic plane appearing in the K3 intersection lattice are examples.

The integral Gram determinant has absolute value one, equivalently the lattice equals its bilinear-form dual. The K3 intersection lattice is an indefinite example.

An integral quadratic lattice that is both an even lattice and a unimodular lattice. The K3 intersection lattice has signature $(3,19)$, while the positive-definite E8 lattice has rank eight.

The positive-definite rank-eight even unimodular lattice can be realized as vectors in ${\mathbb{Z}}^8$ or $(\mathbb{Z}+1/2{)}^8$ whose coordinate sum is even, with the ordinary Euclidean inner product. Negating this form gives $E_8(-1)$ in the K3 intersection lattice.