Spacetime and observer space symmetries in the language of Cartan geometry

We introduce a definition of symmetry generating vector fields on manifolds which are equipped with a first-order reductive Cartan geometry. We apply this definition to a number of physically motivated examples and show that our newly introduced notion of symmetry agrees with the usual notions of symmetry of affine, Riemann-Cartan, Riemannian and Weizenb\"ock geometries, which are conventionally used as spacetime models. Further, we discuss the case of Cartan geometries which can be used to model observer space instead of spacetime. We show which vector fields on an observer space can be interpreted as symmetry generators of an underlying spacetime manifold, and may hence be called "spatio-temporal". We finally apply this construction to Finsler spacetimes and show that symmetry generating vector fields on a Finsler spacetime are indeed in a one-to-one correspondence with spatio-temporal vector fields on its observer space.