But most of ours have a separate clutch pack on each axle side that are operated independently, not just one for the whole diff.
If you look at the cutaway photo above, you can see the two clutch packs. The one at the differential input (where the prop shaft would attach) is the "whole diff" clutch that controls power flow into the rear differential.
On the side of the input clutch pack is the clutch pump. There are two hydraulic lines that exit and go up and over the differential, into a clutch pack on the output side (where an axle shaft would bolt to). This clutch pack is the "eLSD" that controls power flow between the left and right rear wheels.
GM introduced this system with Haldex on Saabs (just before they were shut down, unfortunately). It also received use on the Cadillac SRX (twin to the Saab 9-4x) and numerous Epsilon platform vehicles (Opel Insignia, Buick Regal and Lacrosse, and Cadillac XTS). I believe the XTS is the last remaining vehicle still using this system. Most new GM's use an AWD system from GKN.