Let R be an associative ring with unity 1 and let a, d ∈ R. An element a ∈ R is called invertible along d if there exists unique a d such that a d ad = d = daa d and a d ∈ dR ∩ Rd (see [6, Definition 4]). In this note, we present new characterizations for the existence of a d by clean decompositions of ad and da. As applications, existence criteria for the Drazin inverse and the group inverse are given.