The pilot-in-command was a student on a check flight for a private license. The check pilot was an FAA inspector. The flight was conducted from an airport with runways that were partially covered with snow and ice. The mishap occurred while the pilot was demonstrating a short field takeoff on the third departure. According to the pilot, the aircraft began to veer to the left during rotation. The pilot stated that he then brought the nose down, shut the power off, applied brakes, and the aircraft slid into the snow. The inspector reported that the plane lifted off prematurely, then settled back onto the runway after a very short time. When the plane slid to the left of the centerline, the inspector put her foot on the right rudder pedal. She found that the pilot already had it completely depressed. As the left main gear went into the snow, the aircraft was pulled farther left. It came to rest 6 feet off of the runway with damage to the propeller, left wing, and an engine mount.