Suppose we have a smooth manifold $M$ and an embedded/immersed submanifold $N \subset M$.
As usual, for every $p \in N$, we identify $T_pN$ as $(di)_p(T_pN)$, where $i:N \hookrightarrow M$ is the inclusion.
Now let's say that we have two smooth vector fields $X$ and $Y$ on $M$ which happen to be tangent to $N$ (i.e. $X_p, Y_p \in T_pN, \forall p \in N$).
We know that in this case, the Lie bracket of $X$ and $Y$ in $M$, call it $[X,Y]_M$ is also tangent to $N$.
However, what about their Lie bracket on $N$? By this I mean, taking the restrictions $X^*$ and $Y^*$ of these vector fields to $N$ (i.e. two smooth vector fields $X^*, Y^*$ on $N$ which are $i$-related with $X$ and $Y$, respectively), how would we compute the Lie bracket of $[X^*, Y^*]_N$ in $N$?
It seems "trivial" that $$ [X^*, Y^*]_N = \left. [X, Y]_M \right|_{TN}. $$ However, how would we go about proving this? I don't really know how to start, as it seems just too trivial.
Context: I firstly thought about this when I encountered the vector fields $$ X_1 = z\partial_y - y\partial_z, X_2 = x\partial_z - z\partial_x, X_3 = y\partial_x - x\partial_y, $$ which are smooth vector fields on $\mathbb{R}^3$ but also "happen" to be tangent to the sphere $S^2$. I then wanted to compute the Lie bracket $[X_1, X_2]$ on $S^2$. I just computed it in the classical way in $\mathbb{R}^3$, and then restricted it to $S^2$.