Any primitive/public camp sites around Elk Lake?

Planning on doing the Dix loop with a base camp at one of the tent/lean-to areas along the trail that goes north from the south tip of Elk Lake. On the hike out i'd like to see if there's any places along Elk Lake to pitch a tent. I'll have a kayak with me on the car at the trail head, so water transport is fine. Can you camp on any of those little islands in the lake? Or is this all private property?
thanks!
josh