Planar discrete-subgroup basis

A nonzero discrete additive subgroup of ${\mathbb{R}}^2$ has a basis of one or two real-linearly independent vectors. Choose a shortest nonzero vector $w_1$. If the subgroup is not contained in its line, reduce parallel components modulo $w_1$ and choose a vector $w_2$ of least positive perpendicular height. Such a height exists: heights tending to zero would give infinitely many reduced points in a bounded region, contrary to discreteness. Reducing any group vector modulo $w_2$ leaves height smaller than this minimum, hence height zero, and reducing along $w_1$ then gives an integer multiple of $w_1$.