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Suppose that we have two vector fields $V,W \in TM$ with $M$ a differentiable manifold.

We define the Lie Bracket as $[V,W](f)=V(W(f))-W(V(f))$ and it's easy to show that this bracket is indeed a vector field.

I feel like I'm missing something but then why is not possible to define the Lie Bracket as a tensor taking two vector fields and a covector?

In that case how should I prove this?

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If such a tensor exists, $T(fX,gY)=fgT(X,Y)$. In coordinates, note that $[{\partial \over \partial x_i},{\partial \over \partial x_i}] =0$, we would have $[x_i{\partial\over \partial x_i},{\partial \over \partial x_i}]=0$, but the Lie bracket is ${\partial \over \partial x_i}$

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