Let $(M, g)$ be a Riemannian manifold and $X, Y, Z$ smooth vector fields on $M$. Let $\theta_X$ be the $1$-form defined as $\theta_X(Y) = g(X,Y)$ and let $d\theta_X$ be its exterior derivative. Let $L_X$ denote the Lie derivative. Let $\nabla$ be the Levi-Civita connection, i.e. the unique torsion free connection on $(M,g)$ which is also compatible with the metric. I want to prove that: $$ 2 g(\nabla_Y X, Z) = (L_Xg)(Y,Z) + (d \theta_X)(Y,Z). $$ Is there a quick way to see that?
EDIT: I'm stuck with the $2$-form $d\theta_X$. I don't know how to deal with it. Can I express it just using the metric and the Lie derivative (without working in local coordinates)?