Coordinate horizontal lifts of a principal connection ($H_i={\partial }_i-A_i^a{R}_a,[H_i,H_j]=-F_{ij}^a{R}_a$)

On a coordinate trivialization of a principal bundle, the right-invariant vector fields $R_a(\gamma )=T_a\gamma$ satisfy $[R_a,R_b]=-c^d{}_{ab}{R}_d$. With $A=A_i^a{T}_{a}d{x}^i$, the displayed lifts project to the commuting coordinate fields and contract to zero with the principal connection. Taking their Lie brackets gives the local curvature of a principal connection coefficients $F_{ij}={\partial }_i{A}_j-{\partial }_j{A}_i+[A_i,A_j]$. Thus these lifts commute exactly when the connection is flat. Arbitrary base fields need not commute, and flatness alone does not supply a global coordinate frame or remove holonomy of a connection.