I would like to know if there is any way to calculate the Haar-measure of the Lie group $ SL (2, \mathbb{R}) $ with respect to the $ KAK $-decomposition, where:
for each $ g \in SL (2, \mathbb{R}) $, there are $ k_1, k_2, a \in GL (2, \mathbb{R}) $ such that $ g = k_1 a k_2 $ for
$ k_i = \left (\begin{array}{cc} cos (\theta_i) & -sin (\theta_i) \\ sin (\theta_i) & cos (\theta_i) \end{array} \right) $ for $ i = 1,2 $ and $\theta_i \in \mathbb{R}$.
$ a = \left (\begin{array}{cc} \alpha & 0 \\ 0 & \alpha^{- 1} \end{array} \right) $ for $ \alpha \in \mathbb{R}_{> 0} $.