(“They have no flat sides.”)
If you already “get” this, it’s simple. But it’s worth repeating for all the times I see these ridden incorrectly. Serpentines are curves. They have no flat sides. They touch the track at the appropriate places on the long side but don’t cling to it.
The ones you see in training level and second level begin and end at the middle of the short side, that is at C or A. Once you begin the serpentine, you do not go in the corner. A 3 loop serpentine in the large arena is ridden like 3 halves of 20 m circles. When the serpentine ends at the middle of the short side, you then DO ride into the next (second) corner. This includes training test three when after the walk you begin a serpentine at C. Immediately start the arc of a circle first touching the track again 10 m from the end of the arena.
Loops— like in the canter in first level three—are different. They don’t start and finish on the short side, but rather at the first and last letters on the long side of the ring. This means you would ride through the corner only leaving the track at (for instance) M. Loops are also to be ridden as all curves with the bend of the figure changing as you cross the quarter lines.
This is unlike a “vee” which from time to time has been called for in a test during the free walk: “free walk K – X – H.” Here you would ride two straight lines with the turn at X from one diagonal onto the other.
As you complete your loop at the last letter on the long side of the arena, you then do go into the corner around to C.