I want to understand the following statement on the page 3 of the note written by Martin Hairer: http://www.hairer.org/papers/hormander.pdf on Hormander's theorem.
- Lie bracket $[U, V ]$ is between two vector fields $U$ and $V$ on $\mathbb R^n$ is the vector field defined by $$[U, V ](x) = DV (x) U (x) - DU (x) V (x).$$ If we denote by $A_U$ the first-order differential operator acting on smooth functions $f$ by $A_U f(x) = \langle U(x), \nabla f(x) \rangle$, then we have the identity $$A_{[U, V ]} = [A_U, A_V ].$$
My question is how to derive the above claim. I think I am not even clear about the notions here. For instance, what is the inner product $\langle \cdot, \cdot \rangle$? If we interpret it by $$\langle v, u \rangle = \sum_{i=1}^n v_i u_i, \forall u, v\in \mathbb R^n,$$ $A_U$ is a mapping from $C^\infty(\mathbb R^n, \mathbb R)$ to itself.
As mentioned in the above comments, one wants to show that $$A_{[U, V]} f = A_U A_V f - A_V A_U f, \ \forall f\in C^\infty.$$ For simplicity, we assume $n = 1$. Then, the left hand side is $$LHS = [U, V] f' = (UV' - U'V) f', $$ and right hand side is $$ RHS = U (V f')' - V (U f')' = (UV' - U'V) f'. $$