The formula for the principal curvature form in the Wikipedia article Curvature Form, when applied to the $X, Y$ tangent vectors to the principal bundle $P$, goes on like $$\Omega(X, Y)=d\omega(X, Y)+\frac12[\omega(X), \omega(Y)]$$ where $\omega$ is the connection form with value in the lie algebra of the structure group.
It only depends on the lie bracket of the lie algebra, not on any conventional definition of the wedge product of forms
Having learned that there is no $\frac12$ coefficient, I was about to edit the article when I saw a caveat from the authors "do not remove the $\frac12$ coefficient". They did motivate its presence by a correct calculation made from the definition given in the article Lie-Algebra Valued Forms of Wikipedia where there is a $\frac1{(p+q)!}$ coefficient in front of the definition of the wedge product of two lie-algebras valued forms.
This definition is different from the one I saw in Michor's book Natural Operations in Differential Geometry, Cap. 19, or other textbooks, where the coefficient is $\frac1{p!q!}$. This explain the difference in the formula for the curvature.
What is right? It seems to me that there should be no such a coefficient $\frac12$ and that I should edit the article on Lie-Algebra Valued Forms. Am I right?
Edit 2
Well, the only explanation that makes sense for me is the following: the authors of the article are using an "old" convention for the definition of the exterior derivative of a one form, following there the one given in the Kobayashi & Nomizu book they referred to in a note.
In this book, the convention used put a $\frac1{p+1}$ in front of the definition of the exterior derivative of a $p$-form compare to the "modern" definition.
So, for the connection form , we have a $\frac12$ coefficient in the formula
$$d\omega(X,Y)=\frac12(X.\omega(Y)-Y.\omega(X)-\omega([X,Y]))$$
With this convention, it is easy to see that the formula for the curvature given in the Wikipedia article is right, albeit giving a curvature that is half of the modern definition for it.
Having two competing definitions for such an important and basic operator is a source of confusion and waste of time IMHO. I will definitely go for the "modern" approach.
Thus, I just edited the wikipedia article with a reference to the choice of convention made for the exterior derivative definition, following my answer. It seems so me the safest path of action, since getting rid of the $\frac12$ coefficient would have implied a change in other articles.
Therefore, my remaining questions are now 1/ is my reasoning right? 2/ is there any rationale behind the choice of the Kobayashi convention for the exterior derivative versus the modern one or is it just a matter of convention? In any case, the choices of some wikipedia articles (like Curvature Form, or Lie-Algebra Valued form, etc.) of the Kobayashi convention is confusing when it is not clearly specified.