Yes - but it's more instructive to run the argument backwards

Suppose the

**F**-field, then the 2-form

, is given by a section of the principal bundle on Minkowski spacetime, a manifold recall, whose structure group

is a Lie group. Now define,

*ad hoc*, so to speak, the 1-form

, the vector potential. We will assume, again on the fly, that this is a connection on our bundle.

Then I can show, with some difficulty I grant you, that, by definition of the covariant derivative,

is the curvature of

. Notice that curvature is a Lie algebra-valued (

-valued) 2-form by definition.

So, all is well, since we know that

is a 2-form. And we also know that, by definition,

, the Bianchi identity .

But, if we

*already know* that

then it must be that

, then this requires that our structure group has an algebra with all brackets zero. And if we further require our group to be unitary, then we must choose

, where of course all Lie brackets vanish.

Thus have we derived the necessary (but not sufficient) structure group for "ordinary" electromagnetism.