I got excellent results in rehab’ing a lower back injury from a physiotherapist. From what I saw of the other patients there, he used to use similar regimens for them.
NB the abs are at least as important as the lower back for stability in that area. Also, I was required to do all my exercise with my tailbone tucked under my pelvis, the “pelvic tilt” adopted by many KF’ers.
The exercises went progressively along these lines:
alternate lunges
lying on your back, bend one knee foot flat on floor push up with the bent leg to lift your butt off the floor
standing on a stait or raised beam, one leg squats (only going about an eighth of the way down) - keep the pelvic tilt
bent leg raises, making sure the lower back is pressed into the floor and the pelvis does not move
After a few weeks of this, gradually increasing the number of reps, I was moved on to:
alternate single leg bounds forward, working on baliance and precise foot placement
jumping in place from lunge on one side to the other
leg raises as before with a ball or other soft weights held between the knees (add weight progressively, don’t let that pelvis move)
continue the one leg squats
squats against the wall, holding a ball between the ankles, then the knees
This was a rehab program for injury rather than a program for a healthy person, but it worked really well - I had had years of lower back pain due to an injury on top of some congenital spinal problems, this cleared it up permanently in a few months. I’m doing regular to heavy KF training, including high kicks, and BJJ as a blue belt, and haven’t had back pain since.
But, I emphasise, see a professional for any rehab requirements.