Let $M$ be a smooth manifold and $N$ a submanifold of $M$. Let $X$ and $Y$ be vector field of $N$.
We know that the differentials on $M$ of those vector fields $d_XY$ and $d_YX$ are not necessarily tangent to $N$, and so we need to project them on to $TN$ to define a connection on $N$.
However, looking at the proof in my course that the projected connection is torsion free (for $M=\mathbb R^n$), it seems like the same is not true for the Lie bracket on $M$ defined by $[X,Y]=d_XY-d_YX$ which is tangent to $N$ without needing to be projected. Why is this true?