Basis of a toric lattice

A lattice basis $v_1,\dots ,v_n$ expresses each vector of the toric lattice uniquely as ${\sum }_i{a}_i{v}_i$ with $a_i\in \mathbb{Z}$. Relative to any fixed integral basis, vectors form another lattice basis exactly when their determinant is $\pm 1$. A primitive vector is one whose integer span is saturated; it can be completed to such a basis by the Euclidean algorithm.