Say you have a scalar field defined on a manifold M. Then imagine evaluating the field at a point an infitesimal distance away from x at where X is a illing vector of the manifold. Then
to first order. Now the second term on the rhs is the Lie derivative of a scalar, so
Say now that is not a scalar, say maybe a vector of spinor or tensor. Does the same thing follow, that
Where now is not the appropriate Lie derivative of whatever object is.