Yep. That's all pretty much correct. I'd made soap, in basic chem labs, and I'd tried it, in my kitchen. A quick Google for soap making recipes will give you an answer, for how we separate soap from glycerol. But those "cookbooks" won't tell you -- why it works. But when you, a chemist read it, you might get an insight. Try it and see. You'll be better off, if you get the "A-ha" burst, instead of us just dumping the answer in your lap.