In the proof of the Frobenius theorem , at least the one I saw, eventually we need the fact that
Let $X_1,...,X_k\in \mathfrak{X}(M) $ be vector fields such that $\{X_1|_p,...,X_k|_p\}$ are linearly independent for all $p\in M$. and $[X_i,X_j]=0$ for all $1\leq i,j \leq n$. Then there exists a unique $k$-dimensional foliation $\mathcal{F}$ such that for all $p\in M$ : $T_p L=\langle X_1(p),...,X_k(p)\rangle $ where $L$ is the leaf containing $p$.
Now the idea is to define $L=\{\phi_{X_1}^{t_1}\circ ...\circ \phi_{X_k}^{t_k},t_1,...,t_k\in \mathbb{R}\}$ and this is defined for small enough $t$. Now using the commutativity of the flows of the vector fields it's easy to see that $T_pL=\langle X_1(p),...,X_k(p)\rangle$, my problem is how can one show that is in fact a foliation, how do we find the foliated charts and how do we go about uniqueness ? Any help with this is aprecciated, thanks in advance.