= Solution
A <principal bundle> consists of smooth manifolds $P,B$, a <Lie group> $G$, a smooth surjection $\pi:P\to B$, and a smooth free right action of $G$ on $P$, whose orbits are the fibers. Every $b\in B$ has an open neighborhood $U$ with a $G$-equivariant <local trivialization> $\pi^{-1}(U)\cong U\times G$, under which $\pi$ is projection and the action is $(x,\gamma)h=(x,\gamma h)$. Equivalently a local section $s:U\to P$ writes every fiber point uniquely as $p=s(x)\gamma$. On overlaps, sections differ by smooth transition functions $s_\beta=s_\alpha u_{\alpha\beta}$ satisfying the cocycle identity.
A <principal connection> is a Lie-algebra-valued one-form $\omega$ on $P$ satisfying $R_h^*\omega=\operatorname{Ad}_{h^{-1}}\omega$ and $\omega(\xi^\#)=\xi$ for the <fundamental vector field> $\xi^\#(p)=\left.\frac d{dt}\right|_0p\exp(t\xi)$. Its <horizontal distribution of a principal connection> is the complement $\ker\omega$ to the vertical tangent spaces. \b[The local <gauge potential of a principal connection> is Lie-algebra-valued], specifically $A=s^*\omega$; matrix notation identifies the adjoint action with conjugation. The displayed connection expression is understood on each coordinate trivialization, not as a choice of a global section.
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