Yellow engine icon light on
I have a mini 2012 with 10000 miles. I had a message my gas tank lid was off and then
the engine icon came on and it is yellow. Can the loose gas lid cause the icon for the
engine to become illuminated
Yes it can. It's part of the Evap system to lower emissions, almost all new vehicles have a similar system. To prevent fuel vapors from escaping the fuel tank it is kept under a slight vacuum (old vehicles vented through the caps to atmosphere). A loose cap or leak in any of the Evap components or tubing/hoses will cause this issue.
Yep, the light should go off on it's own in a few days if that was indeed the problem, if you have any doubts it looks like you are still under warranty so you can take it to a dealer if you really want to.
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