The most likely culprit is the oil filter housing gasket.
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https://www.m3post.com/forums/showthread.php?t=1312849
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The S65 engine (E90/E92/E93 M3) oil filter housing gasket (Part # 11427838276) is a common failure point that causes high-pressure oil leaks, often resulting in oil running down the driver's side of the engine. Replacing this rubber seal is intensive, requiring removal of the intake box, belts, and potentially the power steering pump, costing roughly $150-$500+ in labor.
Key Details for Replacement:
Gasket Part Number: 11427838276.
Symptoms: Oil leaking down the left side of the engine, oil on the oil pan, or burning oil smell.
Recommendation: Replace immediately when found to avoid engine damage.
Associated Parts/Tasks: It is recommended to replace the oil cooler gasket at the same time, perform an oil change, and replace the aluminum fasteners.
Common Pitfalls & Tips:
Tight Space: The rear bolt is notoriously hard to reach, requiring a very short T30 bit or a cut-down socket.
Fasteners: Ensure all aluminum fasteners are replaced, as they can stretch.
Cleaning: Thoroughly clean the mounting surface on the engine block to ensure the new gasket seals properly.