The pilot receiving instruction reported that he and the flight instructor were performing transition training in a tailwheel-equipped airplane. After landing to the dry asphalt runway and during the landing roll, the airplane drifted to the left and the pilot applied right rudder and right brake. The airplane corrected to the right and the flight instructor applied left rudder and left brake to straighten the direction of the airplane. Shortly thereafter, the left main landing gear axle failed, the wheel departed, and the airplane continued to travel to the right. The airplane departed the runway to the right and came to rest upright on a grass field. During the runway excursion, the left wing impacted the ground. The two occupants were able to egress from the airplane without further incident. The airplane sustained substantial damage to the left wing. The pilot classified the accident as a “typical tailwheel ground loop caused by the left gear axle failing and the pilot potentially overcorrecting on rollout.” The pilot further reported that the left axle (constructed of hollow aluminum) is an original part from when the airplane was manufactured in 1950.