If you asked Sensei Nishiyama what is the best training for yoko geri, he would say “do yoko geri”, and this is totally true.
Most of the training for a karate person should be karate technique and timing, but at the same token, some supplemental training can help.
We talk about moving from the center and out of optimal posture. If we don’t have a strong and stable center we cannot move effectively from the center and might also hurt our spine, especially when producing big forces. If we don’t move out of optimal posture we won’t fulfill our potentials and have the potential of “repetitive pattern overload” injury, and if our muscles are not balanced we cannot keep a good posture even if we understand what it should be.
Strong abdominals will not necessarily protect your back, people with strong abdominals still have back pain. We must learn to activate the core in proper sequence, smaller, inner muscles should fire before the bigger outer muscles. The smaller, inner muscles, stabilize the spine and protect it while providing an anchor for the bigger, outer muscles, so those bigger muscles can function more optimally. In most of our training, the core should be strengthened while integrated with the rest of the body, not in isolation.
Stretch the tight muscles to prevent imbalance and strain on the spine, such as the hip flexors, the huge iliopsoas muscles are attached to the femur bone and to all five lumbar vertebras, and if tight will create a constant strain on the lumbar spine. Tight hamstrings are also a reason for strain on the spine.
We must constantly improve our posture, with specific postural corrective exercises, a head forward posture create strain on the lumbar spine, and also interfere with force transfer from the body center to the arms.
There are specific exercises for postural correction and learning proper muscles activation sequence that I like to use as part of a warmup.