It's part of why you had the rubber dishwashing gloves get invented: soap used to be harsher, and one of the reasons why is the fact that soaps were made with lye.
Commercial dish soap can be soap, or it could be detergent. Detergent is made differently, and doesn't necessarily involve lye.
> The production of toilet soaps usually entails saponification of triglycerides, which are vegetable or animal oils and fats. An alkaline solution (often lye or sodium hydroxide) induces saponification [...]
i.e. that's the usual process, and it often involves lye. (It elsewhere mentions potassium hydroxide too.)
Anyway my point really wasn't too disagree about how to make soap, I've never done it, it just seemed weird to me to tell someone how they made something and that it's a problem, vs. 'Did you use lye? If so...' or even 'Assuming you used lye, ...'
For example, I think what you call 'commercial dish soap' I would call 'washing up liquid' (never soap, not would it cross my mind if someone else said 'soap') so a slightly different conversation could easily have been based on a wrong assumption about what the created thing was or was used for.
Use dish detergent (commercial dish soap). It's made in a different way, so you can't get un-reacted lye.