TITLE: Non-associative commutative binary operation
QUESTION [33 upvotes]: 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!

REPLY [2 votes]: If $A$ is an abelian group, then the new operation $a * b = - (a+b)$ is commutative but is typically not associative.   (This comes up naturally in the classical chord-tangent law on a cubic curve $C$ in the projective plane: if $a$ and $b$ are two points on the curve, then $a*b$ is the third point of  intersection of the line through $a$ and $b$ with $c$.)