Background: When I was reading Morita's "Geometry of Differential Forms", the author remarked on Page $81$ that
It is easy to see that if a distribution $\mathcal{D}$ on a $C^\infty$ manifold is involutive, then for an arbitrary open submanifold $U$, the restriction $\mathcal{D}|_U$ of $\mathcal{D}$ to $U$ is also involutive.
Question: How can I see it? To prove the author's remark, it suffices to show that every vector field on $U$ belonging to $\mathcal{D}|_U$ agrees on an open neighborhood of $p$ in $U$, $\forall p\in U$, to a vector field on $M$ belonging to $\mathcal{D}$. Is this statement about vector fields true? I think that partition of unity can be used to prove it.
Definition: A distribution $\mathcal{D}$ on a smooth manifold $M$ is involutive if for any two vector fields $X,Y$ on $M$ belonging to $\mathcal{D}$ (i.e., $X_p,Y_p\in \mathcal{D}_p$ for all $p\in M$), the bracket $[X,Y]$ also belongs to $\mathcal{D}$.
You don't even need the full strength of a partition of unity. Assume $X\in\mathfrak{X}(U)$ is a vector field on $U$ everywhere tangent to $\mathcal{D}\vert_U$ and $p\in U$. Take a bump function $\varphi\colon M\rightarrow\mathbb{R}$ such that $\varphi\equiv1$ on a neighborhood of $p$ and $\mathrm{supp}(\varphi)\subseteq U$. Then, $\varphi X$ is a smooth vector field on $M$ that is everywhere tangent to $\mathcal{D}$ (indeed, it is the zero vector field on the open set $M\setminus\mathrm{supp}(\varphi)$ and a scalar multiple of $X$ on the open set $U$, the union of these two is $M$). Clearly, $X$ and $\varphi X$ agree on the neighborhood of $p$ where $\varphi\equiv1$.