I'm looking at the problem of parametrizing a group element $g \in SL(n,\mathbb{R}).$
I think I understand the concept of parametrization $-$ just a change of coordinates, right? But I don't understand the method (too bad, I heard afterwards that there was an lecture-example of parametrising an element $g_{SU}$ of $SU(2)$ to the form $g_{SU} = \begin{pmatrix} x_0 + ix_1 & x_2 + i x_3 \\ -x_2 + ix_3 & x_0 - ix_1 \end{pmatrix}$).
Say, in $SL(n,\mathbb{R})$ for an element $g = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ we only have the constraints that $a, b, c, d$ are real and that $ab - cd = 1$. As a hint, I'm supposed make the parametrization so that I get a constraint for a "pseudosphere" (a hyperplane, more like it?), of the form
$c_0 x_0^2 + c_n x_3^2 + x_1^2 + x_2^2 = c$, where $c_0, c_3, c = \pm 1. \quad (1)$
Q1: From the aforementioned constraints, how am I supposed to deduce the coordinate transforms in analogy to the SU(2) case? I can always try something heuristically similar, like assigning $a \mapsto x_0 + x_1$ and $b \mapsto x_2 + x_3$ and go from there, but what is the method to finding the parametrization?
Q2: And if I manage to find a transformation that gives me the constraint of the form (1) do I know that it is unique? How? Intuitively the problem would make no sense if it wasn't.