An integral lattice is a finite-rank free Abelian group 
equipped with an integer-valued symmetric bilinear form 
. A basis
gives an integral symmetric Gram matrix. The lattice
is nondegenerate when this matrix has nonzero determinant.
Positive definiteness is an additional condition, rather than part of the integral
convention used here (Milnor and Husemoller 1973).
An integral lattice is even when 
is even for every 
, and odd otherwise. It is unimodular when the determinant
of a Gram matrix is 
or 
. For a nondegenerate lattice, this is equivalent to 
, where 
is its dual.
For a nondegenerate integral lattice, the signature 
counts positive and negative squares
of the real extension of the form. A Lorentzian lattice has signature 
, or 
with the opposite sign convention. The even unimodular
Lorentzian lattice denoted 
has signature 
and occurs in the Borisov-Nuer
conjecture.
