# Regarding coordinate and non-coordinate bases

Kontilera
Hello PF members!
I have a problem regarding coordinate and non-coordinate bases.

As I understood from my course in GR, the partial derivatives of a coordiante system always commute:
$$[\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}] = 0 .$$
Which is not necessarly true for non-coordinate bases.

However in Giorgi's book 'Lie Algebras in Particle Physics' he starts out by parameterizing a Lie group G, by a set of N real parameters. (I.e. a coordinate system.)
The he shows that if we taylor expand around the identitiy (for a representation) the we get a set of generators which are the Lie algebra of our Lie group G.

My question then is:
Since he started out by choosing his generators from a coordinate system doesn't this mean that he will find:
$$[X_a, X_b] = 0 \quad?$$

// Kontilera