2013 Drake Smith - Please do not use or reproduce this elsewhere.  Feel free to link to it though.

1985 & Earlier Fuel Level Sender

Wiring:

Brown

Ground

Green/White

12V+ to Fuel Pump

Brown/White

4L Low Fuel Lamp in Instrument Cluster

Brown/Black

7L Low Fuel Lamp in Instrument Cluster

How It Works:

Fuel Pump Power: The fuel pump only receives power when the engine is turning.

Fuel Level Sensors: The 7 liter and 4 liter indicators in the instrument cluster are triggered by separate sensors which are at different heights in the fuel tank. When they are submerged, the fuel around them keeps them cool. When the fuel gets low and they are exposed to air, they heat up which completes a circuit that sends a current to the instrument cluster.  I've never tested or measured them but, given that the  wires' main color is brown and from the instrument cluster wiring diagram, I'm pretty sure that it's a ground signal.

Flaky Connector:

As these bikes age, it is not unheard of for the connector from the main wiring harness to the fuel level sender to deteriorate and cause the fuel pump to get intermittent or no power.  Fortunately, BMW was nice enough to provide you with a spare connector. 

If you look in the relay box you will find a round white four terminal connector which is not used. (It's for the optional alarm.)  This is the same style of connector that is used for the fuel level sender so you can use it to replace a flaky fuel level sender connector.

2013 Drake Smith - Please do not use or reproduce this elsewhere.  Feel free to link to it though.