When I got my diff fluid done (pre LSD) I gave the mechanic a US quart (946ml) and it was literally the perfect amount (came to the bottom of the fill plug) despite BMW TIS saying its got a 0.85L fill quantity.
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As for the auto box I am not sure, but for my manual I used 2 US quarts (1.9L) and there was nearly half of one of the bottles left over. I think from what I've read though that the auto box has a much higher drain and fill capacity because when its drained you will also drain the torque converter (with the engine off). TIS seems to indicate there is a 9L capacity including the torque converter.
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Hope this helps.