Corollary of Leibniz rule for Lie bracket

Statement

This is an identity that uses the Leibniz rule to measure the failure of the Lie bracket operation from being $C^{\mathrm{\infty }}$-linear.

Let $X,Y$ be smooth vector fields on a differential manifold $M$ and $f$ be in $C^{\mathrm{\infty }}(M)$. We then have:

$f[X,Y]=[fX,Y]+(Yf)X$