TITLE: Can a single nontrivial identity imply associativity and commutativity?
QUESTION [6 upvotes]: As noted in this old question it's easy to see that no single identity is equivalent to the conjunction of the commutative and associative laws.
Question. In the language of a binary operation, is there a single nontrivial identity which implies both the commutative law and the associative law?
By "nontrivial" I mean that it does not imply that the operation is constant.
I suppose the answer is well known to experts on universal algebra. If possible, please give the explanation "for dummies", as any technicalities are likely to go over my head.

REPLY [9 votes]: Yes.
According to Theorem $1$ of 
Minimal Identities for Boolean Groups
N. S. MENDELSOHN AND R. PADMANABHAN
JOURNAL OF ALGEBRA 34, 451-457 (1975)
a structure $\langle A, \star\rangle$ satisfies the single identity
$(x\star y)\star (x\star (z\star y)) \approx z$
if and only if the structure is an abelian
group of exponent $2$. 
[The fact that such an identity exists follows from work of Higman and Neumann from 1952, but the paper I cite gives short identities.]