Engines
Engine oil is the most difficult to spec, and you'll get a lot of discussion about this. You can obviously use any motor oil you want, but you should be using an oil designed for use 1) with flat tappets, and 2) NO catalytic converter. In the USA, engine oils are rated by the American Petroleum Inst. About 1969 the API instituted a 2 letter code system, with the first letter standing for the type engine; "S" for gasoline, and "C" for diesel.

These old engines require (were designed for) an oil API rating of SF or SG. About the only way to get that these days is to buy a "racing oil", such as Valvoline or Castrol. These oils still have the ZDDP "zinc package" to protect your cam shaft and tappets. Unfortunately, the federal DOT took these metal compounds out to help protect catalytic converters in cars (the place most oils being sold are used). Those newer, more common, oils are labeled API SM or SN.

You could simply buy SN rated oil and add the ZDDP as an additive, but that is actually the more expensive way to go. And there is no real proof this actually works. You may also find some oils labeled "motorcycle oils" and certain synthetics which have this zinc protection, but the key again is the API rating. Typically though, a thorough scrutiny of the on-line MSDS (material safety data sheet) will show you which oils have the most ZDDP, and that is typically the "racing oils".

Diesel engines never had the catalytic converter, so they still retain the ZDDP. In winter you can save money by buying 15W40 diesel oil, API rated something like CJ-4. (Here you must understand that there are 2-stroke diesels and 4-stroke diesels, so you definitely need to see that numeral "4" on the label.) Being less expensive and easier to find at discount stores, diesel 15W40 is also a good break-in oil or short term oil to use after a long storage period.

Personally, I use the non-synthetic Valvoline 20W50 VR1 most of the year in GA. This is typically available at all local auto parts stores by the quart. The only place I see this sold by the gallon (the less expensive choice) is at Auto Zone.

[Along with the above information, your Airhead has other related engine oil requirements. To be fully informed, you should also research topics concerning: engine oil warm up, oil filter o-ring stack up, and oil filter change intervals.]

Power Transmission
The 1) gearbox, 2) drive shaft, and 3) final drive all use the same weight extreme pressure "gear oil". These come in a wide range of weights from 70/90 to 75/140. Based on personal success with the Valvoline VR1, I've chosen Valvoline gear oils for use in a 1979 R100 RT and been very happy.

Generally speaking, synthetic gear oils offer much lower friction rates (as evidenced by lower running temperatures), and display noticeably better shifting characteristics. However, as with all synthetics, they also (again, generally speaking) tend to leak past oil seals at a much higher rate. However, IMHO the difference in performance is to great to ignore, and one should at least experiment.

My experience has shown that the amount of "seal softeners" in synthetics differs widely in from brand to brand. My brother's Oilhead came with Mobil-1 (the seller was very proud of this), but it completely covered the rear wheel within 300 miles. The simple act of draining and refilling with Valvoline synthetic gear oil stopped the leak entirely, without changing the oil seal. The same Valvoline synthetic gear oil has been in the 3 transmission compartments of my RT without leaking and with the added improvement in shifting characteristics now for ~3000 miles. I'm simply delighted.

Closing
Whether you agree or disagree with this, one thing everyone agrees on is that oils are much cheaper than BMW parts. The better practice is to always to change your oils early and often. If you feel like you're not wringing every penny of value out of your oils by doing this, then you can always place the drained oils in your lawn mower or work bench oil can !

I'll also add that 1) my findings may differ from others, 2) were discovered purely by accident, not by any specific research, and 3) although I'm a degreed engineer, I am not now, nor ever have been employed by any oil company or petroleum product distributor.

Hope this helps.