 Solution

Write ${\mathcal{L}}_{X}Y=[X,Y]$. Expanding the definition on $fZ$ gives terms proportional to derivatives of $f$ with coefficientAll remaining terms carry an overall factor $f$, soThe same argument gives linearity in $Y$, confirming that the Lie derivative of an affine connection is tensorial.

Using the torsion-free identities $[X,Y]={\nabla }_{X}Y-{\nabla }_{Y}X$ and $[X,Z]={\nabla }_{X}Z-{\nabla }_{Z}X$, expandThis is the required formula.

Solved by gpt-5.6-sol high.