The general Frobenius Theorem stating that

Let $u_1,\dots,u_k$ be $k$ smooth linearly independent vector field on $M$. Let
$$
W=\operatorname{Span}(u_1,\cdots,u_k)
$$
Then $[u_i,u_j]\in W$ for any $i,j$ if and only if there exist foliation by $k$ dimension hypersurface tangent to $M$.

To my understanding,

there exist foliation by $k$ dimension hypersurface tangent to $M$.

means there is a cooridnate $(w_1,\cdots,w_{n-k},x_1,\cdots,x_k)$ such that
$$
u_i=\frac{\partial}{\partial x_i}
$$

I know the proof of the special case where $k=2$. To prove the general case, there is a hint saying using induction on $k$. However, I am not clear how to commit the induction. I tried to consider $[u_{k-1},u_k]$, but we only have
$$
[u_{k-1},u_k]\in\operatorname{Span}(u_1,\cdots,u_k)
$$
rather than
$$
[u_{k-1},u_k]\in\operatorname{Span}(u_{k-1},u_k)
$$
So I cannot perform the simpler case. Could anyone help?

This post imported from StackExchange Mathematics at 2014-09-24 21:03 (UTC), posted by SE-user hxhxhx88