Just did mine and did this writeup.

HID for euro lights (http://members.roadfly.com/faisal_in_laurel/E31 HID Install for Euro Lights.pdf)

Great instructions but I miss one point:
Whenever I did a HID conversion on E32 or E34 (never did an E31 so far) I needed to fool the LKM because it would normally indicate non working headlights. How did you solve this issue on your installation or is the digital ballast "accepted" as a bulb by the system?

In my 850 it did not give me an error. But in my 840 it did and was resolved by using a Ressistor pack from Bavauto.com part # 87705 (for 2000 M5) inline with the system that sat between the Ballast and the Igniter.

What was your method of fooling the LKM?

http://www.bavauto.com//assets/product_images/lighting/87705.jpg