Motor oil: Valvoline 10W30 VR1 synthetic
Gearbox lube: 80W-90 GL4 gear oil was originally recommended. The lube needs to be compatible with the conflicting needs of limited slip differentials AND synchronizer rings. There is very little brass, bronze, or copper in the gearbox, that is a minor concern. The synchronizers are made of iron (unless they've been replaced with brass). The ring and pinion are spiral-bevel cut gears, not hypoid gears.
PennGrade 1 Classic Multi-Purpose GL-4 SAE 80W90
Castrol Axle Limited Slip 80W-90 Gear Oil (GL-5)
Although Castrol gearbox oil is rated GL-5 nobody reports any problems associated with its use. The consensus is that what's most important is changing the oil annually.
Redline MT-90 75W90 GL-4 Gear Oil
Redline 75W90NS GL-5 Gear Oil
Some guys have good luck with synthetic oils in the gearbox, others find the synthetics tend to leak (drip) from all the seams between castings and from the seals, sometimes the leaks stopped dripping after a period of time, once the seals had a chance to soften and the gaskets a chance to swell, etc. When the drips haven’t stopped owners have returned to petroleum based oils, and the leaks stopped leaking.
One person has reported that Lloyd Butfoy advised him against synthetic lube, whereas Les Gray is adamant regarding the use of Redline 75W90NS GL-5 Gear Oil. These are both gentlemen whose advice I respect.