In the Compatible connections section of the Wikipedia article Connection form, it says for a connection on vector bundle $(E,p,M)$ with structure group $G$ to be called compatible with the $G$-structure, the following must hold locally:
$$Γ(γ)_0^te_α(γ(0))=\sum_βe_β(γ(t))g_α^β(t)$$
where $γ$ is a curve on $M$, $e_α$ and $e_β$ are basis local sections in a local frame $\mathbf e$ on $E$, $g_α^β$ as a whole is a $G$-valued smooth function of $t$.
and it says differentiation of the right at $t=0$ gives
$$\sum_βe_βω_α^β(\dot γ(0))$$
where $ω_α^β$ as a whole is a $\mathfrak g$-valued form.
How is the differentiation defined and done?
The problem is that the value of the function to be differentiated at different point is not subtractable, which commonly means parallel transport is needed, but the equation here is already about parallel transport so it seems to be a dead loop.
Also, can the differential be done on each summand or can it only be done on the sum as a whole?