# Lie Derivative of one-form: an identity

1. Mar 22, 2015

### Quality Cheese

1. The problem statement, all variables and given/known data
I am trying to prove an identity for the Lie derivative of a smooth one-form. The identity is: for X, Y smooth vector fields, alpha a smooth one-form, we have:
$$L_{[X, Y]}\alpha = [L_X, L_Y]\alpha$$ For anyone familiar with the book, this is exercise 5.26 in the first edition of Nakahara: Geometry, Topology, and Physics.

2. Relevant equations
I am given the identity: for X, Y smooth vector fields, alpha a smooth one-form,
$$(L_X\alpha)(Y)=L_X(\alpha(Y))-\alpha([X, Y])$$ ([X, Y] is the Lie Bracket of the vector fields X and Y, and $$L_XY=[X, Y])$$

3. The attempt at a solution
I keep trying to expand the lie derivatives and cancel terms but I think I am missing a property of the lie derivative; maybe I'm messing up the Leibniz rule or something? Any help would be greatly appreciated!

2. Mar 27, 2015