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A lattice basis $v_1,\dots ,v_n$ of a toric lattice $N$ expresses each element uniquely as ${\sum }_i{a}_i{v}_i$ with $a_i\in \mathbb{Z}$. Relative to a fixed integral basis, the new vectors form a lattice basis exactly when the matrix of their coordinates has determinant $\pm 1$.

Being a real basis is weaker. The vectors $(1,0),(1,2)$ are linearly independent over the real numbers, but their integer combinations have even second coordinate and do not fill ${\mathbb{Z}}^2$. Their determinant is two.

A primitive lattice vector is nonzero and generates a saturated sublattice. In coordinates its entries have greatest common divisor one. The Euclidean algorithm completes it to a lattice basis. The nonzero condition excludes the zero vector, whose zero subgroup is saturated but which cannot be a basis vector.^[ref]