A matrix $\begin{bmatrix}w&x\\y&z\end{bmatrix}\in\mathbb Z^{2\times 2}$ is unimodular if $$|wz-xy|=1$$ holds.

Is there a parametrization of such matrices with $|w||z|-xy=1$ $$w,z<0<\max(y|z|,|w|x)<\frac{|w||z|+xy}2?$$

Additionally I would prefer to have the constraint $$\max(|w|,|z|,x,y)\leq2\min(|w|,|z|,x,y)$$ which I think has to hold automatically if a parametrization exists.