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!

Adding a little to @DietrichBurde's answer, Jordan algebras (characterized by $xy=yx$ and a limited associativity $(xy)(xx)=x(y(xx))$ are a reasonably natural not-fully-associative class of algebras. The standard Jordan algebras are made from associative algebras with multiplication $\cdot$ by defining $xy=x\cdot y+y\cdot x$. For non-commutative algebras, this gives a non-associative Jordan operation. For example, such notions are useful in harmonic analysis on the associated cones: for the associative algebra of $n$-by-$n$ real matrices, the associated Jordan algebra structure is useful in studying the cone of positive-definite $n$-by-$n$ matrices. Jordan's original motivation was from observables in physics, but/and, again, self-adjoint things are prominent.
