Parabolic element of PSL2(R) ($|\mathrm{tr}A|=2,A\ne \pm I$)

A nonidentity real Möbius transformation represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$ with $|\mathrm{tr}A|=2$. It has one ideal boundary fixed point and is conjugate to a nonzero translation on the Poincare half-plane model.