The pilot probably pushed the "wheel" forward...hard...as soon at the oxygen warning bell rang.
Powered airplanes go "down" by reducing the throttle, not pointing the nose down. Loss of cabin pressure calls for an immediate descent. Airliners generally will maintain cabin pressure unless the leak is severe.
Small jets usually cruise at 40,000 feet plus but a pressurization leak will quickly cause death due to the small cabin (unless emergency oxygen is applied immediately).