I just purchased a differential rear axle that I hope to modify for my cycle kart.  I say “hope” because the axle is actually from a zero-turn-radius walk behind commercial lawn mower.  It is too short and I am concerned that it won’t be able to withstand the speed and abuse, but if it works it will work well.  I also anticipate creating an attachment to lock the differential if I ever find myself using the cycle kart in looser dirt where a live axle would perform better.