SOAP

If it doesn't have lye in it, it's not soap. Some producers of 'homemade' soap, understanding peoples' aversion to lye, list sodium hydroxide or caustic soda as an ingredient instead of lye. Sodium hydroxide is lye. Caustic soda is lye.
Many commercial soaps do not list lye or sodium hydroxide as an ingredient. That's because the 'cleansing bar' you buy does not contain lye so is not soap - it's detergent. They may add lanolin or other softening agents including glycerin which is a by product of soap making. But without lye, it just isn’t soap. When you make soap at home, glycerin is created as a by-product, making the soap wonderfully soothing for the skin.