I know that every coset representative $g\in SL(2,\mathbb{C})$ for $SL(2,\mathbb{C})/SU(2)$ can be chosen of the form
$ g = \left( \begin{array}{cc} \sqrt{t} & \frac{z}{\sqrt{t}}\\ 0 & \frac{1}{\sqrt{t}} \end{array} \right), z\in \mathbb{C}, t>0, z=x+iy $ and then $g$ is mapped onto $z+kt$ in the quaternionic upper half plane
$\mathcal{H}^c = \{z+kt = x+iy +kt | x,y\in \mathbb{R}, t>0\}$
the elements are equal to quaternions from
$ \mathbb{H} = \mathbb{R} \bigoplus \mathbb{R} i \bigoplus \mathbb{R} j \bigoplus \mathbb{R} k$
with the $j$-coordinate equal to zero. An exercise says: Show that the invariant arc length on $\mathcal{H}^c$ is given by
$ds ^2 = (dx^2 +dy^2 +dt^2 )t^{-2}$
This would proof that $SL(2,\mathbb{C})/SU(2)$ can be identified with the hyperbolic $3$-space, wouldn't it? Do I have to start with a Riemannian metric on $SL(2,\mathbb{C})/SU(2)$? What is the metric on $SL(2,\mathbb{C})$?
I would appreciate any explanation of basic knowledge that is behind a possible solution here, since I am not very familiar with quaternions or Lie groups. Thank you in advance!