Coordinate transformation on the Christoffel tensor Xcoba

I have to compute the Christoffel Symbols on a basis which is not the coordinate basis of the chart in mathematica.
I should do this contraction:

[CapitalGamma][-a, -b, c] = e[c, -[Gamma]] e[[Alpha],a] (e[[Beta], b] [CapitalGamma][[Gamma], -[Alpha], -[Beta]] + partial_[Alpha] e[[Gamma], b])

is it possible to make mathematica to do it by herself?

with DefBasis and BasisChange and AlllComponents I’ve been able to introduce the new basis and to write the metric in this basis. Now I’m dealing with computing the Christoffel symbols (which are no more symmetric) in the new metric but I Don’t know how to make mathematica to compute them contracting the old Christoffel with the basis elements. I think, as it had developed the correct rules by the previous commands, the program should be able to compute them autonomously, do you know?
Thank you