Check the color and shape of the fuel pump relay and compare it to the horn relay. Not sure on the E53, but on the E46 the horn and fuel pump relay are the same and you can just swap them. The horn relay rarely is used while the fuel pump relay is used every time the vehicle is started and is energized the entire time the engine is running.

I would doubt the horn relay on most 10 year old vehicles has seen more a few dozen cycles.

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2008 4.8i Black with Tabacco. Black Y-spoke wheels and shadowline trim.