Sounds like the ignition pulse is going to the lamp but then the ballast is not providing the correct voltage and current to keep the arc sustained and the lamp then goes out again. Most likely a ballast fault. Swapping them is te easiest way to check them. Testing needs great care as the lamp ignition startup operating voltage and associated current can kill as it is up to 23kV depending on type.

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Me:
Current: 2011 E71 40D Vermilion Red/Beige Nappa

Wifey:
Current: 2012 R80 Countryman SDX Green/Cream