A method of operating a four-stroke gasoline direct-injection controlled auto-ignition combustion engine includes opening both the intake and exhaust valves during terminal portions of the expansion strokes and initial portions of the contraction strokes, injecting fuel directly into the combustion chamber for mixing with retained gases and igniting the fuel near the ends of the contraction strokes. In the process, combustion gases are expanded to produce power during mayor portions of the expansion strokes, combusted gases are blown down into the exhaust outlet and the air inlet and are partially redrawn into the cylinder with fresh air during the terminal portions of the expansion strokes so the air charges are heated by the hot exhaust gases. Portions of the charges re-expelled and theremaining portions of the charges and injected fuel are compressed for ignition of the dilute fuel/ air and exhaust gas mixture. Substantial reductions of NOx emissions result from the method.