Since the metric tensor is a symmetric bilinear form, it feels like I should be able to expand a vector like this:
$ Z = g(Z,e_k)e^k $
Where the e_k together spans the space (maybe they need to be orthogonal wrt g, maybe g(e_k,e_k) need to be 1)
Now, from https://en.wikipedia.org/wiki/Fundamental_theorem_of_Riemannian_geometry:
$ g(\nabla_X Y,Z) = \frac{1}{2}(\partial_X(g(Y,Z)) + \partial_Y(g(X,Z)) -\partial_Z(g(X,Y)) + g([X,Y],Z) - g([X,Z],Y) - g([Y,Z],X)) $
Where ( I guess )
$ [X,Y](f) \equiv X(Y(f)) - Y(X(f))\equiv[\partial_X,\partial_Y](f)\\ \partial_X(F)(p) \equiv \lim_{t\to 0}t^{-1}[F(p+tX(p))-F(p)] $
So, I wonder if I can find coefficients for $\nabla_X Y$ by doing this:
$\nabla_X Y = g(\nabla_X Y,e_k)e^k = \frac{1}{2}(\partial_X(g(Y,e_k)) + \partial_Y(g(X,e_k)) -\partial_{e_k}(g(X,Y)) + g([X,Y],e_k) - g([X,e_k],Y) - g([Y,e_k],X))e^k$