Scan the ECU for codes and see what it says. I take it you got no other lights besides EML?

Is it idling badly or just running rough underload, when it runs?

I have gotten EML for several reasons, and they are common for pre 2004 E53. Usually you will get a P series code from the OBDII though. Check it out.

EML clears when you turn off the car, but the code that triggers it will not have cleared. If the CPU detects the same conditions at start up again, it may throw the EML again. This can happen intermittantly until you rememdy the underlying problem.

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2001 E53 3.0 5pd