Sufficient discreteness conditions for $2$-generator subgroups in $\mathrm{PSL}(2,\mathbb C)$

Every element in $\mathrm{PSL}(2,\mathbb C)$ is elliptic, parabolic, or loxodromic. For the groups generated by two elliptic elements, sufficient discreteness conditions were obtained by Gehring, Maclachlan, Martin, and Rasskazov. In this article we establish sufficient discreteness conditions for the groups generated by two loxodromic elements and the groups generated by a loxodromic element and an elliptic element.