Let $\nabla$ be a symmetric, linear connection on a smooth manifold $X$.

If $p \in X$ is any point, on a normal chart for $\nabla$ around $p$ it holds: $$ \Gamma_{ij}^k (p) = 0 \ , $$ where $\Gamma_{ij}^k$ denote the Christoffel symbols on those coordinates.

I am wondering whether a similar statement holds for principal connections.

To be precise, let $G$ be a Lie group and let $P_0 := G \times X \to X$ be the trivial principal $G$-bundle. Consider a principal connection on it, defined by a 1-form $\alpha$ on $P_0$ with values on the Lie algebra $\mathfrak{g}$ of $G$.

If $(g,x)\in P_0$ is any point, is it possible to find a connection $\bar{\alpha}$ isomorphic to $\alpha$, whose value $\bar{\alpha}_{(g,x)} $ at that point is zero?