I'm quite new to Physics and I was having a look for the first time to the Standard model. I'm not sure if the mechanism that I'm describing is directly from Weyl or from others but what I found quite interesting was this:
1- We have a Lagrangian Density which has a global symmetry expressed by a Lie Group $G$ (let's think about a Lie Group such as $SU(2)$, $U(1)$, $SU(3)$ etc...);
2- I try to construct a Lagrangian Density which has the same symmetry but locally and not globally;
3- To do so I introduce an affine connection;
4- I replace all the derivatives in the Lagrangian Density with the Covariant derivatives;
5- I add an "ad hoc" term to the original Lagrangian a term that directly involves this connection in the Lagrangian Density;
6- with these manipulations I've done (with some effort and some variants from group to group) the Lagrangian density is now locally invariant for the Lie Group I was starting with.
From this point on the Standard Model follows interpreting the proprieties of this Lagrangian. Now, the "magical" part that is interesting me is something that can very may well be evident to an expert of Riemannian Geometry.
My question is: why that doing such a construction the structure constant of the Lie Algebra corresponding to the original Lie Group appear in the Curvature Tensor related to the affine connection I introduced? Does this depends to the specific form of the Lagrangian or is something which express some deeper geometrical fact that I'm missing?