generalized to the most general action with a single scalar field which leads to second order differential equations [@Deffayet:2011gz], keeping the theory free from the Ostrogradsky instability [@Woodard:2006nt]. This model has been shown to be Horndeski’s original work [@Kobayashi:2011nu]. These models have not only attracted the attention of cosmologists but also the community of black hole physics, because it constitutes a simple field to check various no-go theorems.

In this paper, we will derive the full stability conditions of a black hole solution subjected to a linear odd-parity gravitational perturbation. This work has already been approached by other authors [@Kobayashi:2012kh]. In their paper, they have expanded the Horndeski action to second order in odd-parity perturbations and succeeded in identifying the master variable, they obtained the no-ghost and no-gradient instability conditions (see [@Kobayashi:2012kh] for more details). All these results have been recovered in our paper, but we have derived the correct generalized Regge-Wheeler equation[^1] which is fundamental in the study of the stability of black holes and quasinormal modes. We have performed the full analysis of the stability condition, by expanding the action to second order of perturbations, obtain the no-ghost and Laplacian stability conditions and derive the correct generalized Regge-Wheeler 