Small Engines (Lawn Mowers, etc.): Briggs and Straton Engine Popping, head gasket, briggs and straton

Question
I have a Briggs and Straton 22 hp Intek Model 40777-1031-E1 on an John Deere L120.  The motor is popping/backfiring when idling.  It seems to go away when mowing, but it is hard to tell.  I have adjusted the valves, rebuilt the carb (soaked the carb, blew out all the parts with compressed air and reassembled) and changed/gapped the spark plugs.  Could the problem be the shear key?  The head gasket doesn't appear to be leaking/damaged and the pushrods/valves appear to be fine.

Answer
Matt
If the shear key is off just a little that will causing the timing to be off,What you need to do is check the following:

Problem: Engine backfires.
Possible causes:
Carburetor set too lean.
Magneto is not timed correctly.
Valves are sticking.

Then if you need to pull the flywheel just make sure the key way is in the correct spot,