I'm reading the Wikipedia page on Levi-Civita connections, and there's a step that I don't quite follow in one of the derivations. I was wondering if someone could help.
Suppose $X$, $Y$, and $Z$ are vector fields on a manifold $M$, $\nabla$ is an affine connection on $M$ and $g$ is a Riemannian metric on $M$.
How can one show that $X (g(Y, Z)) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$ using only the definition of $\nabla$? Do we require compatibility with the metric?
Edit: one of the comments has pointed out that this is the definition of compatibility with the metric. I don't see how to derive this relationship from the definition that I've seen, i.e. $\nabla g = 0$. Thanks!
Edit 2: Question is now a dupe of Equivalent definition of metric compatibility for a connection: what does $\nabla g$ mean?