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Is there a specific product rule or something one must follow when applying the lie bracket/ commutator to two vector fields such that one of them is multiplied by a function and added to another vector field? This is the expression given in my textbook but I don't see how:

[fX+Z,Y] = f[X,Y] + [Z,Y] - (Yf)Xg

I don't see where the third term on the right hand side comes from.

I'd really appreciate some help on this because I'm self-learning differential geometry for a research project and almost all my doubts revolve around my not understanding how lie brackets work. So any help will be appreciated.

Thanks!

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# Action of Lie Brackets on vector fields multiplied by functions

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