We prove that any interpolatory Lie group subdivision scheme based on combining a linear interpolatory subdivision scheme with the log– exp adaption to Lie-group-valued data in Ur Rahman et al. (2005, Multiscale Model. Simul., 4, 1201–1232) produces parameterized curves on the Lie group that are as smooth as the smoothness of —no matter how smooth is. We present both an extrinsic proof and an intrinsic proof. We discuss two variations of our main result. (i) We illustrate how smoothness equivalence can break down in a variant of the original log– exp scheme. (ii) We show that the main result of this paper can be easily extended to a multivariate setting.