The Lie derivative of a tensor is a tensor of the same rank and type. But it is connection independent meaning it can be expressed in terms of covariant or partial derivatives. Since the Strong Equivalence Principle (SEP) requires the connection, at a point on a Lorentzian manifold, to vanish, spacetime is locally Minkowskian. At that point, the connection in Lie derivative when expressed in terms of its covariant derivatives, vanishes as well. So it seems the Lie derivative always satisfies the SEP and local Lorentz invariance. However, in Gravity and Strings by Ortin, it says on P374: the Lie derivative is not covariant under local Lorentz transformations and its action on Lorentz tensors is frame-dependent. So, does the Lie derivative of a tensor satisfy the SEP and local Lorentz invariance or not?