Because Pilates requires you to press your abs toward your spine, you don't want to allow your lower belly to round and press out as air comes into your lungs. You also don't want that abdominal lock to force you to breathe shallowly. To breathe correctly, you must expand your rib cage, primarily through your midback.
Your lower back contains numerous muscles. One group, called the erector spinae, attaches to your spinal column at different points along your back, allowing you to bend it forward and backward and from side to side. Toning these muscles helps prevent back pain, and it also tightens and firms your entire lower back.