As mentioned in the link in the op, all soap is lye soap. All the big commercial soaps add colors and scents, but remove most or all the glycerin that is produced by the saponification of fats or oils by the lye. My daughter makes and sells soap but she adds essential oils to add odor to most of them but does not remove the glycerin. Adds menthol to one as a sinus soap, activated carbon to another to make acne soap. I have not asked her, but she might be willing to make a plain soap with no additives.
https://naturalstatesoap.com/ Not give as a solicitation to buy, but as a reference for some versions that are available. Since the covid pandemic started, she has not been making near so much.
When I worked in the kitchen at the college cafateria bakery, we used lye soap for washing the baking pans and pie pans.