I am trying to show that the Borel measure $\frac{\mathrm{d}a\mathrm{d}b\mathrm{d}c}{|a|}$ on $SL(2,\mathbb{R})$ is right invariant.
Let $\left( \begin{array}{cc} r & s \\ t &u \end{array} \right)\in SL(2,\mathbb{R})$ and let $f: G \rightarrow \mathbb{R}$ be measurable.
$$\int_{SL(2,\mathbb{R})}f\left( \left( \begin{array}{cc} r & s \\ t &u \end{array} \right) \left( \begin{array}{cc} a & b \\ c &d \end{array} \right) \right) \frac{\mathrm{d}a\mathrm{d}b\mathrm{d}c}{|a|} = \int_{SL(2,\mathbb{R})}f \left( \left( \begin{array}{cc} ar+bt & as+bu \\ cr+dt & cs+du \end{array} \right) \right) \frac{\mathrm{d}a\mathrm{d}b\mathrm{d}c}{|a|}.$$
Now I want to apply a change of variables. So set $n = ar+bt$, $m = as+bu$ and $k = cr+dt$. Then one gets $|Jac| = |r|$ and $$\int_{SL(2,\mathbb{R})}f \left( \left( \begin{array}{cc} n & m \\ k & {*} \end{array} \right) \right) \frac{\mathrm{d}n\mathrm{d}m\mathrm{d}k}{|r| |nu - mt|}$$ where $*$ is just a place holder for a longer term. At this point I am stuck. Is the idea to just calculate right? Did I make a mistake so far? And if not, can somebody give a hint how to proceed from here?