Many attribution studies of precipitation extreme events have attempted to estimate the thermodynamic contribution (linked to temperature changes) and the dynamic contribution (linked to the atmospheric circulation). Those studies are based on statistical decompositions of atmospheric fields, and essentially focus on the horizontal motion of the atmosphere. This paper proposes a framework that decomposes those terms from first physical principles, which include the vertical atmospheric motion that has often been overlooked. The goal is to take into account the driving processes of the extreme event. We revisit a recent example of extreme precipitation that was extensively investigated through its relation with the atmospheric circulation. We find that although the horizontal motion plays a minor (but important) role, the vertical motion yields a dominating contribution to the event that is larger than the thermodynamic contribution. This analysis quantifies the processes leading to high winter precipitation rates, and can be extended for further attribution studies.