The horizontal plane encompasses movements primarily in the forward-back and right-left directions.  When the body is unified, movements in the horizontal plane also manifest a downward grinding force.  To understand the downward grinding, we can draw specific points from the 13 Points: suction on the dantian and sternum, tucking of the ribs, and wrapping the elbows to the ground.  Maintaining these points engages the muscles of the shoulder girdle and abdominal muscles; the end result is stabilization of the arms relative to the shoulder and the clamping of the shoulders down towards the hips (or if you prefer the clamping of the sternum down to the dantian).

The most common mistake in manifesting the horizontal plane energy is pushing down instead of grinding down.  Pushing down increases activation of the chest and arms, involves leaning and rolling the shoulders forward, and directs the primary motion downwards out of the horizontal plane.  By comparison, the grinding power comes aligning the structure to land at the horizontal. With the structure aligned correctly, the force created by tucking the ribs, wrapping the elbows, opening the mingmen, etc. knock the bones in an arced path that will sustain a downward grinding power without having break neutral position just to push down. From the upper hand position, the grinding power generates a downward engaging force at the point of contact that allows pulling and pushing from the point of contact.  While not the only application of the horizontal plane energy, it is the clearest demonstration of the horizontal plane in action:

Text revised 08/01/2022 for more precise wording.