Commutators, and Christoffel symbols in a non holonomic basis I have a frame that varies along a curve $\gamma$ : the frame consists in the tangent vector of the curve plus some constant non orthogonal vectors.
I need to compute $\nabla_{\dot\gamma}{\dot\gamma}$. Apparently, the Christoffel symbols in that case are computed with :

$$\omega^i{}_{k\ell}=\frac{1}{2}g^{im} \left( g_{mk,\ell} +
 g_{m\ell,k} - g_{k\ell,m} + c_{mk\ell}+c_{m\ell k} - c_{k\ell m} 
 \right)\,$$
where $c_{k\ell m}=g_{mp} {c_{k\ell}}^p\ $ are the commutation
  coefficients of the basis; that is, $[\mathbf{u}_k,\mathbf{u}_\ell] = c_{k\ell}{}^m \mathbf{u}_m\,\ $

(wiki)
I have several questions regarding the commutation coefficients:


*

*I read that the commutator is defined by $[X,Y]=\nabla_X Y - \nabla_Y X$ : isn't it a catch 22? how is it possible to compute that commutator, knowing that the connection $\nabla$ is obtained by the Christoffel symbols that uses the commutator that uses the connection that uses [...] ?

*For one of my applications, my tangent space is made of symmetric matrices (and the manifold has a fancy metric $g$) : in that case, is the commutator only defined by $[X,Y]=X*Y-Y*X$ with $*$ the standard matrix multiplication ?

*For another application, my tangent space are vectors in $R^N$ and my space is Euclidean. Of course, in that case there are probably easier ways to handle that, but I'd still like to understand what would the commutator be in that case.

*In the definition $[\mathbf{u}_k,\mathbf{u}_\ell] = c_{k\ell}{}^m \mathbf{u}_m\,\ $ , is-it necessary to solve a linear system to get the coefficients $c_{k\ell}{}^m$ or can I just project $[\mathbf{u}_k,\mathbf{u}_\ell]$ on $\mathbf{u}_m$ using the metric $g$ (my basis is not orthogonal, so I guess the linear system is needed?).


Thanks!
 A: This "commutator" $[X,Y]$ is in fact the Lie bracket of two vector fields $X$ and $Y$ seen as derivations on the algebra $C^\infty(M)$ of smooth functions on the manifold $M$.
This Lie bracket is appropriately defined as an operator on $C^\infty(M)$ given by
$$
[X,Y]f := X(Y\,f) - Y(X\,f) \tag{1}
$$
where $f \in C^\infty(M)$.
It turns out to be a derivation again, and so represents a vector field $[X,Y]$.
Now the trick is that one can show that
$$
[X,Y] = \nabla_X Y - \nabla_Y X \tag{2}
$$
for any torsion-free connection $\nabla$, in particular, for the Levi-Civita connection that you are using in your calculations.
This is an example of a Lie algebra with the bracket defined by (1). Another example that you have mentioned is the algebra of square matrices or order $n$ with the bracket given by the commutator of the matrices. It is not necessarily related to the first example, of course.
It must be emphasized that in definition (1) we do not use any metric or connection at all, only a smooth structure on the manifold, that is the choice of charts that we may have chosen to identify $M$ locally with pieces of $\mathbb{R}^n$.
Simply put, just compute the directional derivatives!
Remark. The formula that you give in the quotation is the expression for the Christoffel symbols of the Levi-Civita connection. You can prove it as an easy consequence of the so-called Koszul formula.
(All references are in the appropriate articles in Wikipedia)
