What is meant when one says that one has chosen a basis of fields on the manifold with ``anolonomy"?

I get the feeling that it is a choice of basis with non-trivial structure constants say $C^{k}_{ij}$

Now in some papers people seem to state that once such a basis is chosen one can write the Christoffel symbols in terms of the structure constants by the following equation,

I am writing the equation with all indices lowered (as written in the original paper)

$\Gamma_{ijk} = \frac{1}{2}(C_{ijk} - C_{ikj} - C_{jki})$

I suppose they are assuming that the connection is chosen to be torsion free since the above equation satisfies torsion free-ness condition.

But just being torsion free doesn't seem to be enough to derive the above relationship.

This relationship is seen in the papers in the context of choosing a veirbein on homogeneous spaces equipped with a Riemannian metric.

I would like to know from where and how does the above equation come.

In general it is surely impossible that the structure constants determine the connection!

constantsin the general case. $\endgroup$