Dear MO\_World,

I'm hoping someone can point me towards a reference for something. I have an invertible $2\times 2$ matrix, $A$, with real entries such that for both of the rows, the entries are rationally independent (this ensures that $A\mathbb Z^2$ only intersects the coordinate axes at the origin).

What I want is a pair of generators of the lattice, $u$ and $v$, with $u$ belonging to the first quadrant and $v$ to the fourth quadrant. 

I have a proof that I'm not entirely happy with using continued fraction convergents, but as it's going in an already-long paper, I'd love to have a self-contained reference for this.

Does anyone have any suggestions?