Lost in translation: The Abelian affine connection (in the coincident gauge)

The simplest, i.e. the Abelian, i.e. the commutative, i.e. the integrable, i.e. the flat and torsion-free, i.e. the symmetric teleparallel affine connection has been considered in many recent works in the literature. Such an affine connection is characterized by the property that it can be vanished by a general coordinate transformation, by fixing the so-called coincident gauge. This paper focuses on the subtleties involved in the applications of the coincident gauge.