Lye would only be used in the soponification of oil in hard or bar soap. Sodium hydroxide is used for soft soap. So if that's something you're looking for, I'd recommend Dr. Bronzer's Sal Suds. It's specially made for dishes and laundry as the reaction is 1:1. This is in contrast to bath products (ie bar soap or liquid soap, not like dove but like ones you could get at the farmer's market, etc), which do something called super fatting. Super fatting is an out of balance reaction, leaving some unsuponified left over oil for moisturizing. Sometimes, laundry bars and KOH soft soap can even have a reaction that would leave a very small bit of lye or Sodium hydroxide unreacted for cleaning purposes, only sometimes. That's why it's not recommended to use them in the shower as they can be drying on the skin.

I realize you're referring to detergents in your post, but I thought I'd clear that up. I make soap. Lol.

Dr. Bronzer's Sal Suds is pretty great and it has a very subtle pine smell that doesn't stick around.

edit: clarification