Coordinate fields, Lie bracket of

Question

Why $[\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}]=0$ where

$$[X,Y]=(X^i\frac{\partial Y^j}{\partial x^i}-Y^i\frac{\partial X^j}{\partial x^i})\frac{\partial}{\partial x^j}$$

is the Lie bracket in coordinates with Einstein's summation convention understood.

I'm looking for the simplest argument possible.

Answer 1

Hint: The coefficients of $\frac{\partial}{\partial x^j}$ are constants, so the partial derivatives are zero.

Answer 2

${\partial\over{\partial x^i}}$ is the vector field whose value at the point $x$ is $e_i$ where $(e_1,...,e_n)$ is the canonical basis of $\mathbb{R}^n$ so its partial derivatives are zero since it is constant.