SCHWEITZER 2009 Problem 7

Let $H$ be an arbitrary subgroup of the diffeomorphism group $\mathsf{Diff}^\infty(M)$ of a differentiable manifold $M$ . We say that an $\mathcal C^\infty$ -vector field $X$ is weakly tangent to the group $H$ , if there exists a positive integer $k$ and a $\mathcal C^\infty$ -differentiable map $\varphi \mathrel{: } \mathord{]} - \varepsilon,\varepsilon\mathord{[}^k\times M\to M$ such that (i) for fixed $t_1,\dots,t_k$ the map $$\varphi_{t_1,\dots,t_k} : x\in M\mapsto \varphi(t_1,\dots,t_k,x)$$ is a diffeomorphism of $M$ , and $\varphi_{t_1,\dots,t_k}\in H$ ; (ii) $\varphi_{t_1,\dots,t_k}\in H = \mathsf{Id}$ whenever $t_j = 0$ for some $1\leq j\leq k$ ; (iii) for any $\mathcal C^\infty$ -function $f: M\to \mathbb R$ $$X f = \left.\frac {\partial^k(f\circ\varphi_{t_1,\dots,t_k})}{\partial t_1\dots\partial t_k}\right|_{(t_1,\dots,t_k) = (0,\dots,0)}.$$ Prove, that the commutators of $\mathcal C^\infty$ -vector fields that are weakly tangent to $H\subset \textsf{Diff}^\infty(M)$ are also weakly tangent to $H$ .