Is there a builtin function to expand covariant derivatives in terms of the metric (xAct, xCoba)

Question

Firstly, I realize that this is somewhat easy to do with replacement rules as a one off. However, I see that in my work this is something that I may like to use occasionally without much effort. What I would like in math notation is to go from the form:

\begin{equation} \nabla_\alpha u_\nu = \partial_\alpha u_\nu - \Gamma^\mu_{\ \alpha \nu} u_\mu = \partial_\alpha u_\nu - \frac12 (g^\mu_{\ \ \alpha,\nu} + g^\mu_{\ \ \nu,\alpha} - g^{\ \ ,\mu}_{\nu\alpha}) u_\mu \end{equation} In mostly xAct notation with some pseudo code at the end I was hoping for something that would do the following:

DefManifold[M,4,{a,b,c,d,e,f,i,j,k}]
DefChart[coords,M,{0,1,2,3},{v[],\[Xi][],\[Theta][],\[Nu][]}]
DefMetric[-1,metricg[-a,-b], CD, PrintAs->"g", SymbolOfCovD->{";","D"}]
DefTensor[u[-a],M]
eqn =  (u[-a] metricg[a,b]CD[-b][u[-c]])metricg[c,d](u[-e] metricg[e,f]*CD[-f][u[-d]])
CDtoMetric[eqn]
(* Some result similar to what I have written above *)

I'm still searching the different manuals I have downloaded and if I find the answer I will reply with it at such a time.

Answer 1

As Jose explained in the description the correct solution added to my pseudo-code looks like:

DefManifold[M,4,{a,b,c,d,e,f,i,j,k}]
DefChart[coords,M,{0,1,2,3},{v[],\[Xi][],\[Theta][],\[Nu][]}]
DefMetric[-1,metricg[-a,-b], CD, PrintAs->"g", SymbolOfCovD->{";","D"}]
DefTensor[u[-a],M]
eqn =  (u[-a] metricg[a,b]CD[-b][u[-c]])metricg[c,d](u[-e] metricg[e,f]*CD[-f][u[-d]])
ScreenDollarIndices[
 ChristoffelToGradMetric[CovDToChristoffel[eqn]]]

Seems to give the correct result.