A matrix $\begin{bmatrix}w&x\\y&z\end{bmatrix}$ is unimodular if $w,x,y,z\in\mathbb Z$ and $|wz-xy|=1$ holds.

Is there a parametrization of such matrices with $\max(2y|z|,2|w|x)<(wz+xy)$, $w,z<0$ and $\max(|w|,|z|,x,y)\leq2\min(|w|,|z|,x,y)$?