Do not confuse with Cartan lemma.
Also called Cartan identity, Cartan homotopy formula or Cartan magic formula
$$ \mathcal{L}_{\mathbf{X}}=i_{\mathbf{X}} \circ d+d \circ i_{\mathbf{X}} $$ where for a $p$-form $\omega$ is the $p-1$-form$$ i_X(\omega)(Y_1,Y_2, \ldots)=\omega(X,Y_1,Y_2, \ldots) $$If we take as a definition
$$ i_X(f)=0 $$for a function $f$ then we have the know fact:
$$ \mathcal{L}_{\mathbf{X}}(f)=i_{\mathbf{X}}(df) $$Symbolically it can be denoted
$$ \left\{d, \iota_{X}\right\} \omega:=\mathcal{L}_{X} \omega=d\left(\iota_{X} \omega\right)+\iota_{X} d \omega. $$See formulas for Lie derivative, exterior derivatives, bracket, interior product
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Author of the notes: Antonio J. Pan-Collantes
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