Non-associative commutative binary operation Is there an example of a non-associative, commutative binary operation?
What about a non-associative, commutative binary operation with identity and inverses?
The only example of a non-associative binary operation I have in mind is the commutator/Lie bracket. But it is not commutative!

The simplest example of a nonassociative commutative binary operation (but lacking an identity element) is the two-element structure $\{a,b\}$ with $aa=b$ and $ab=ba=bb=a;$ note that $a=bb=(aa)b\ne a(ab)=aa=b.$ This is the NAND (or NOR) operation of propositional logic, where $a=\text{true}$ and $b=\text{false}$ (or vice versa). See this answer or this one.
