Why is $igF_{\mu\nu}:=[D_\mu,D_\nu]$? Asking from a mathematical perspective (as opposed to a physical one):
Why is the field strength tensor defined to be the commutator of covariant derivatives? I understand the general intuition behind it (the Lie Bracket gives the difference in “flow” in directions, and the covariant derivative tells how the field changes), but I’m looking for a rigorously mathematical perspective.
I have experience with some differential geometry and Lie Algebras.
 A: I think for now I will answer with an example: You can generalize all this by working on principal bundles, connection 1-forms, local gauges and chosing representations accordingly.
Let $M=\mathbb{R}^{n}$, $G=SU(2)$ with Lie algebra $\mathfrak{su}(2)$ and let $\phi:M \to \mathbb{C}^2$ be a smooth function. $SU(2)$ acts by right-multiplication on $\mathbb{C}$. Then let us chose a "connection $1$-form"/gauge potential of the form $A_i=t^aA^a_i$ (may depend on $x\in M$) and define the covariant derivative to be
$$
D_i=\partial_i+A_i
$$
Now, let us calculate
\begin{align*}
[D_i,D_j]\phi &=\partial_i\partial_j \phi-\partial_j\partial_i \phi+A_iA_j\phi-A_jA_i\phi + \partial_i A_j \phi-\partial_jA_i \\
&=[A_i,A_j]\phi+ \partial_i A_j \phi-\partial_jA_i \\
&=F_{ij}\phi
\end{align*}
This works for any smooth $\phi$, since partial derivatives commute. The extra factor of $ig$ is usually a consequence of conventions.
The major simplifications we made are
1)We have a globally defined gauge potential
2)The codomain of $\phi$ is a nice, well known space with a simple representation of $G$ (instead of e.g. a Lie algebra)
3)We use "global coordinates", one of the major advantages of working on $\mathbb{R}^n$ and having a global gauge potential
4)Every object has an interpretation without using sections of this and that bundle.
If you were to work on (associated) vector bundles $E$ equipped with the a covariant derivative $\nabla$, the defining equation for the field strength $F_{\nabla}$ would looks like this:
$$
d_{\nabla}d_{\nabla}\phi=F_{\nabla}\phi
$$
for all section $\phi \in \Gamma(E)$.
