I'm having trouble dealing with the following question : what is the isomety group of $\mathbf{PSL}_2(\mathbb{R})$ viewed as a Lie group with its Killing form ? For the record, its Killing form is the push forward of the determinant on $\mathbf{SL}_2(\mathbb{R})$. I'm almost sure that the component of the identity is $\mathbf{PSL}_2(\mathbb{R}) \times \mathbf{PSL}_2(\mathbb{R})$ acting by right and left multiplication, but i think i don't know enough theory to show it.
Thank you for looking at my problem