Lie Bracket explicit computation

Question

I'm at the beginning of learning about Lie Brackets, and the book I'm working out of explicitly calculates $Zf = [X,Y]f$ (where $X,Y$ are vector fields) as follows: \begin{align*} Zf & = [X,Y]f \\ & = (XY-YX)f \\ & = X(Yf)-Y(Xf) \\ & = X(Y^a\partial_af)-Y(X^a\partial_a f) \\ & = X^b\partial_b (Y^a\partial_a f)-Y^b\partial_b(X^a\partial_a f) \\ & \stackrel{!}{=} (X^b\partial_bY^a-Y^b\partial_bX^a)\partial_af -X^aY^b(\partial_b\partial_af-\partial_a\partial_b f) \end{align*} At ! I'm unsure of how the second term of this is derived. I know I have to use chain rule on $\partial_b(Y^a\partial_a f)$ and the corresponding part of the other term, but when I do this I get: \begin{align*} X^b\partial_b(Y^a\partial_a f)-Y^b\partial_b(X^a\partial_a f) & = X^b(\partial_b Y^a\partial_a f+Y^a\partial_b\partial_a f)-Y^b(\partial_b X^a\partial_a f+X^a\partial_b\partial_a f) \\ & = (X^b\partial_b Y^a-Y^b\partial_b X^a)\partial_a f+X^bY^b\partial_b\partial_af-Y^bX^a\partial_b\partial_a f \end{align*} If I have that $$X^bY^a\partial_b\partial_a f-Y^bX^a\partial_b\partial_a f = -X^aY^b(\partial_b\partial_a f-\partial_a\partial_b f)$$ I'm done. This equality isn't clear to me though, and I'd appreciate an explaination of it. It's also possible I misapplied chain rule or something silly.

Answer 1

Note that \begin{align} X^bY^a\partial_b\partial_a f-Y^bX^a\partial_b\partial_a f &\\ &= X^aY^b\partial_a\partial_b f-X^aY^b\partial_b\partial_a f &\\ &= X^aY^b(\partial_a\partial_b f-\partial_b\partial_a f) &\\ &= -X^aY^b(\partial_b\partial_a f-\partial_a\partial_b. f)\end{align} where we have used

$$X^bY^a\partial_b\partial_a f=X^aY^b\partial_a\partial_b f$$ by changing indices.