I have been trying to do the computations to prove that the Lie brackets $[X,Y]$ are independent of coordinate chart for a while now, but I can´t seem to put the things together.
Let $X|_U = X^i\frac{\partial }{\partial x^i}$ and $Y|_U = Y^i\frac{\partial }{\partial x^i}$ be vector fields via a coordinate chart $(U,\phi =(x^1,\dots,x^n))$ and $X|_V = \tilde X^i\frac{\partial }{\partial y^i}$ and $Y|_V = \tilde Y^i\frac{\partial }{\partial y^i}$ vector fields in coordinate chart $(V,\psi=(y^1,\dots,y^n))$ .
I would like to prove the following identity.
$$[X,Y]|_U = \sum_{i,j} \left(X^i \frac{\partial Y^j}{\partial x^i} - Y^i\frac{\partial X^j}{\partial x^i}\right)\frac{\partial}{\partial x^i} = \sum_{i,j} \left( \tilde X^i \frac{\partial \tilde Y^j}{\partial y^i} - \tilde Y^i\frac{\partial \tilde X^j}{\partial y^i}\right)\frac{\partial}{\partial y^i} = [X,Y]|_V $$
I am aware that the identity $\frac{\partial}{\partial x^i} = \sum_{i} \frac{\partial y^j}{\partial x^i} \frac{\partial}{\partial y^j}$ is usefull.
I am aware as well that there are some questions in MSE that are pointing to this topic, but no one is quite elucidative, my difficulty falls in doing the right manipulations.