Normal subgroups of the Special Linear Group What are some normal subgroups of SL$(2, \mathbb{R})$? 
I tried to check SO$(2, \mathbb{R})$, UT$(2, \mathbb{R})$, linear algebraic group and some scalar and diagonal matrices, but still couldn't come up with any. So can anyone give me an idea to continue on, please? 

${\rm{SL}}_2(\mathbb{R})$ is a simple Lie group, so there are no connected normal subgroups.
It's only proper normal subgroup is $\{I,-I\}$
