© 2018, Springer-Verlag GmbH Germany, part of Springer Nature. We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for operators of the form ∇2ψ+ L(x, ψ, ∇ ψ) which are non-decreasing in ψ.