'soft soap' is not soft soap!

["Mary W. Ballard" <BallardM@xxxxxxxxxxxx>]

My Problem:
As everyone on the list probably knows, "soap" refers to an anionic carboxylate made by the saponification (savon = soap) of glycerides, or lye or caustic + fat/oils boiled produce soap. If you use caustic (50%) you get hard soap--the bars or squares we use; if you use lye (potassium hydroxide) you get soft soap.
So I was thinking the "liquid gels" were soft soaps but the other day, I noticed the chaffed condition of my hands--went to look at the ingredients (listed in order, by law, from major component to minor component): water, Sodium laureth sulfate, sodium lauryl sulfate, DEA (diethanolamine), sodium sulfate, cocamidopropyl betaine, citric acid, sodium chloride (i.e. salt), fragrance, DMDM (dimethyldimethyl?) hydanton, tetrasodium EDTA.
In other words, the 'soft soap gel' is largely a sodium dodecyl sulfate--Orvus WA Paste with a little betaine thrown in (mild surfactant: fatty acid+diamine forms a quat, + carboxylic acid has a amide at one end, the anionic at the other). I checked the pH & it's around 7.0, but I'm wondering, since I usually wear gloves when I degrease/clean with SDS, what am I doing having this Ivory Liquid Gel  on my kitchen counter!
Question: 
Is there another brand that really is a proper soft soap? I would appreciate others looking at their 'soft soap' containers & letting us know! Thanks in advance, Mary Ballard