$\mathbf{P}$ is a $4\times4$ matrix, in the form $$\mathbf{P} =\begin{bmatrix}\boldsymbol{A \quad \bar{B}} \\ B\quad \bar{A}\end{bmatrix} $$ where $\boldsymbol{A,B}$ are $2\times2$ complex valued matrices, where $\mathbf{P}$ has the special property $\det{\mathbf{P}} = 1.$ So this means
$$\boldsymbol{A\bar{A} - B\bar{B}} = 1.$$
I'm looking for a way to parameterise the matrices $\boldsymbol{A,B}$ using hyperbolic functions, namely using the property $\cosh^2{x}-\sinh^2{x} = 1$ where $x\in[0,\infty).$
Would appreciate any help here.