i use 75w70 valvoline LSD oil in stock LSD starlet transaxles.
other well known brand off the shelf oils do the job well.
you need a mineral based oil for the stock LSD as it is a viscous unit. the aftermarket oils such as Redline mt90 etc are too thin for the coupling and results arent as good often creating a biger gap between when the LSD engages and spins the drivers side shaft which shortens the units life