On the properties of forward and backward shifts of vector fields

. The paper investigates some properties of recently defined forward and backward shifts of vector fields. The main purpose of the paper is to show that the forward and backward shifts of vector fields commute with the Lie bracket operator and with some commonly used system transformations. The latter include, for example, classical and parametrized state transformations as well as static and dynamic state feedbacks. These properties become important when studying control problems involving such transformations.