E53 headlights are technically self leveling rather than adaptive.

The arm should be pointing in the direction of the wheel. It is in the correct position. The arm is secured at a particular location so the headlights level as intended. It appears there is a lighter area outbound from where the arm is currently secured. That may be where the connection should be located.

It is normal that the lights move up and down when the key is first turned on. There is just the one sensor so both headlight adjust in unison.

I suggest you add year and model of your X5 in your signature. It is important to getting the correct input.

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X5 4.6 2002 Black Sap, Black interior. 2013 X5M Melbourne Red, Bamboo interior
Dallas