 Loxodromic Möbius transformation

A Möbius transformation with two distinct fixed points is conjugate to $z\mapsto az$. It is loxodromic when $|a|\ne 1$; on three-dimensional hyperbolic space it translates along the axis joining its fixed points and rotates around that axis by $\arg a$. The positive-real multiplier case has no rotation and is also called hyperbolic. When listing elliptic, parabolic, hyperbolic and loxodromic as separate types, loxodromic refers to the remaining case with nonzero rotation. A determinant-one matrix with nonreal trace has reciprocal eigenvalues that are neither real nor of modulus one, so it supplies this latter case.