I saw this simple form of Koszul formula on a book: $$2\ g(\nabla_XY,Z) = \mathcal{L}_Yg(X,Z) + (d\theta_Y)(X,Z)$$ where $\theta_Y$ is the one-form $g(Y,\cdot)$.
It is equivalent to the more commonly seen version since: $$\mathcal{L}_Yg(X,Z) = Yg(X,Z) - g([Y,Z],X)-g(Z, [Y, X])$$ and $$d\theta_Y(X,Z) = Xg(Y,Z) - Zg(X,Y)-g([X,Z],Y)$$
I can follow the proof that Christoffel symbols and this formula are equivalent. But I still don't know how to interpret this. I googled and found that Lie derivative of Riemannian metric is related to strain tensor, is there a geometrical or physical interpretation of this formula?
First, it can be viewed as a decomposition : Here $L_Yg$ is symmetric and $d\theta_Y$ is skewsymmetric (And if $Y$ is a killing, i.e., $L_Yg=0$ or it generates isometries, it shows $(\nabla_XY,Z)$)