A stuck injector will cause a Yuuuge cloud of white smoke, so that's definitely a possibility. The injectors on these vehicles are known to leak, though not like you're describing. As suggested, you need to be able to read the BMW codes. I would think you would be getting misfire codes on the cylinder with the faulty injector (if that's the issue).

AM.

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E70 2010 3.0 M57 AT Titanium Silver with Black Leather
E53 2004 3.0 M54 AT Toledo Blue with Gray Leather
E83 2008 3.0 N52 AT Silver Gray Metallic with Black Vinyl
E46 1999 2.8 M52TU 5MT Black with Tan Leather 250K+ miles (Sold, but not forgotten)