What ingredients in soap are not vegan?
I'm not asking if soaps that are labeled "vegan" are vegan. That's obvious.
I live in a very small town in South America. Actually I have to travel 20 km down a very bumpy road by bicycle to get to a very small town. There are no fancy stores here with soap that are marked "vegan".
Of course, I know that animal fats are not vegan, but I do not think that the bottom-shelf cheap soaps use animal fat (that's a fancy and expensive ingredient).
But what other ingredients in soap might not be vegan? What should I look-for in soap to make sure it's cruelty-free?
In general, are cheap soaps vegan?