Classification of the conjugacy classes of $\widetilde{\mathrm{S}\mathrm{L}}(2,\mathbb{R})$

In this note, we classify the conjugacy classes of ${\widetilde{\mathrm{S}\mathrm{L}}}_2(\mathbb{R})$, the universal covering group of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{R})$. For any non-central element $\alpha \in {\widetilde{\mathrm{S}\mathrm{L}}}_2(\mathbb{R})$, we show that its conjugacy class may be determined by three invariants: (i) Trace: the trace (valued in the set of positive real numbers ${\mathbb{R}}_{+}$) of its image $\bar{\alpha }$ in ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{R})$; (ii) Direction type: the sign behavior of the induced self-homeomorphism of $\mathbb{R}$ determined by the lifting ${\widetilde{\mathrm{S}\mathrm{L}}}_2(\mathbb{R})\curvearrowright \mathbb{R}$ of the action ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{R})\curvearrowright {\mathbb{S}}^1$; (iii) The function ${\mathrm{\ell }}^{\mathrm{♯}}$: a conjugacy invariant length function introduced by S. Mochizuki [Res. Math. Sci. 3 (2016), 3:6].