# Lie Bracket is not a tensor

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?

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 $${\partial \over \partial x_i}$$