= Coordinate horizontal lifts of a principal connection
{title2=$H_i=\partial_i-A_i^aR_a,\quad[H_i,H_j]=-F_{ij}^aR_a$}
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 $A=A_i^aT_a\,dx^i$, 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 $F_{ij}=\partial_iA_j-\partial_jA_i+[A_i,A_j]$. 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>.
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