Yes.

Any name brand synthetic that meets the API spec is fine, in the weight that BMW recommend. Usually that is 5w-30.

Mobil 1 or Castrol Syntec (the BMW product, rebranded) are both fine. I wouldn't use Syntec Blend, as it is not a full synthetic.

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2007 X3 3.0si, 6 MT, Premium, White

Retired:
2008 535i, 6 MT, M Sport, Premium, Space Grey
2003 X5 3.0 Steptronic, Premium, Titanium Silver
2002 325xi 5 MT, Steel Grey
2004 Z4 3.0 Premium, Sport, SMG, Maldives Blue