I'm trying to understand some Yang-Mills and Chern-Simons theory but I'm getting tripped up by some of the mathematics.
I'm confused about the exterior covariant derivative of Lie algebra valued k-forms on a principal bundle P. In particular, I understand the derivation done in section 2.2.2 of https://empg.maths.ed.ac.uk/Activities/GT/Lect2.pdf, but I'm failing to generalize to k-forms (Exercise 2.5 of the same section) and understand the case where V is the Lie algebra of G where the wedge is replaced by a commutator.
Also, there seem to be different conventions(?) for the coefficient in front of the second term of the exterior covariant derivative. I sometimes see a factor of 1/2 (for example in section 3.2.2 of https://empg.maths.ed.ac.uk/Activities/GT/Lect3.pdf) and other places without it. I believe my understanding has to do with commutators and wedge products of Lie algebra valued forms, but I could be mistaken. Could someone explain and perhaps identify my misunderstandings?
(I've looked in Nakahara, but his section on connections on a principal bundle didn't quite address my question.)
EDIT: I think I've managed to narrow down my confusion. In the first set of notes (Lecture 2), they derive an equation for the exterior covariant derivative of the connection form (proposition 2.1) as well as the exterior covariant derivative of a vector valued form (exercise 2.5). My understanding is that the result of proposition 2.1 should be a special case of exercise 2.5, with the Lie algebra action being that of conjugation (commutator), but there's a factor of 1/2 on the former. Can somebody explain this point to me?