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Let $X$ and $Y$ two vector field on the manifold $M$ (dim($M$)= $m$). Show that the Lie bracket $[X, Y] := XY - YX $ is a vector field on $M$.

I tried to compute using local coordinates, so for $X = \sum_{i=1}^{m} X_i \frac{\partial}{\partial x_i}$ and $Y = \sum_{i=1}^{m} Y_i \frac{\partial}{\partial x_i}$ I obtain after computation that \begin{align} [X, Y] = \sum_{i=1}^{m} \left( \sum_{j=1}^{m} X_j \frac{\partial Y_i}{\partial x_j} - Y_j \frac{\partial X_i}{\partial x_j} \right) \frac{\partial}{\partial x_i} \end{align}

But how can I conclude now that the Lie bracket $[X,Y]$ is a vector field on $M$? Thanks in advance!

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  • $\begingroup$ This is a typical homework. See for example here. So follow the hints given there. $\endgroup$ – Dietrich Burde Oct 7 '18 at 19:37
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Assuming everything is smooth:

$\textbf{Lemma}.$ Let $M$ be a smooth manifold. The map $D: C^{\infty}(M) \to C^{\infty}(M)$ is a derivation if and only if, $Df = Xf$ for some smooth vector field $X$.

In particular, this says that derivations of $C^{\infty}(M)$ can be identified with smooth vector fields. So, it suffices to show that the Lie bracket is a derivation of $C^{\infty}(M)$.

$\textbf{Sketch.}$ $$ \begin{align*} [X,Y](fg) &= X(Y(fg) - Y(X(fg)) \\ &= X(fYg + gYf) - Y(fXg + gXf) \\ &= \ldots \end{align*} $$

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