Let $M$ be a smooth manifold, and $X,Y\in\TTT(M)$ smooth vector fields on $M$ . Let $\Theta:\UUU\rightarrow M$ be the flow of $X$ , where $\UUU\subseteq \R\times M$ is an open neighborhood of $\gbra{0}\times M$ . We make use of the following notation: $$\UUU^p=\gbra{t\in\R\,|\,(t,p)\in\UUU},\ \ \forall p\in M,$$ $$\UUU_t=\gbra{p\in M\,|\,(t,p)\in\UUU},\ \ \forall t\in\R,$$ and we introduce the auxiliary maps $\theta_t:\UUU_t\rightarrow M$ and $\theta^p:\UUU^p\rightarrow M$ defined as $$\Theta(t,p)=\theta_t(p)=\theta^p(t),\ \ \forall (t,p)\in\UUU.$$

The Lie derivative of $Y$ along $X$ is the vector field $\LLL_XY\in\TTT(M)$ defined by $$(\LLL_XY)_p=\left. \frac{d}{dt} \cbra{ d(\theta_{-t})_{\theta_t(p)} (Y_{\theta_t(p)}) } \right|_{t=0} =\lim_{t\rightarrow0}\frac{d(\theta_{-t})_{\theta_t(p)} (Y_{\theta_t(p)}) - Y_p}{t},\ \ \forall p\in M,$$ where $d(\theta_{-t})_{\theta_t(p)}\in\mathrm{Hom}(T_{\theta_{t}(p)}M,T_pM)$ if the push-forward of $\theta_{-t}$ , i.e. $$d(\theta_{-t})_{\theta_t(p)}(v)(f)=v(f\circ\theta_{-t}),\ \ \ \forall v\in T_{\theta_{-t}(p)}M,\ f\in C^\infty(p).$$

The following result is not immediate at all.