BMW did not restrict use of petroleum and synthetic oils, stating only to use a GL5 oil, & that it be a Brand Name (BMW-speak for "quality ").
While straight grades are listed in the Owner's Handbook, the only listed multigrade oil is 80W90.
Shifting smoothly--preloading the shift lever and why. I am putting 5 links at the beginning of this article to help with understanding.
MORE throwout bearing information is here: folks have problems understanding, let alone visualizing, how BMW transmissions operate.
My present recommendation is that, if you want to, you may use a synthetic gear oil of good quality.I have no objection to it being used in the driveshaft (of those models using oil there).In fairness, I will also note here that one experienced transmission rebuilder, Tom Cutter, my words here, believes synthetic gear oil in your Airhead is not good, leading to unspecified WEAR. I recommended that you do use them unless: you live in an area where the temperature that you start the bike at will be ~100F or more (that also means the un-started engine temperature generally), and that you will be riding fast &/or with heavy loading, or pulling a sidecar or trailer ...in very hot weather ... There is an exception; some transmissions will stiffen-up gear changing when the oil is hot. There are oils available that replace the "90" with 120 or even 145. Link #5: has never waivered in its Airhead motorcycles' transmission oil recommendations, and this includes the last Airheads. My article is VASTLY more complete, particularly about 5 speed transmission problems.I think they are now good enough for me to recommend them in general of dino oils.In some instances seals & the synthetic oils may not be perfectly compatible. The article does describe the broken clutch lever pin problems, & you will find the the exact part numbers & information in my own for your Airhead. Duane's article does not cover a LOT of what I do & certainly does not cover all my testing, nor how to go about any ideas of overhauling a transmission, shimming it, nor ANYthing about the 17.5 gear angle change, etc. Airhead shifts similarly except that the shift levers are moved by a flat plate cam. Some information not covered in my own transmission article, below, that you are reading. I have some nitpicking with some of the information, mostly not worth relating here.