A great way to train deep core " horizontal plane" is unilateral movements.
Important to keep the spine without rotation while the legs or arms or both moving unilaterally, one at a time or at the same time but opposite arm and leg. Check if you noted any difference from one site to other, if noted make on week side ( where you feel pelvic or spine movements ) more reps.
The exercises "face down", are great to train back and front oblique systems.
Global stabilize you need to train in 3 plans ether as deep ones but it can be bilateral. The exercises as an "AB curl" in the sagittal plane involve the spine moves forward to engage obliques.
Small spine movements in the frontal plane will engage side portions of the oblique and some very important back muscles to keep the lower back healthy. Try to avoid moves back and forward or rotations, make pure side bends.
Long-time professional materials tell personal trainers to not rotate it's bad for the back but finally realized that we need to train those muscles as we do twists in daily life every day. This move ad important as any others, so let's make them work to give move nutrients to spinal joints. Avoid any side bends while you rotate. The ribcage should be pelvic as slender.