A classification of endo-commutative curled algebras of dimension 2 over a nontrivial field

An endo-commutative algebra is a nonassociative algebra in which the square mapping preserves multiplication. In this paper, we give a complete classification of endo-commutative curled algebras of dimension 2 over an arbitrary nontrivial field, where a curled algebra satisfies the condition that the square of any element is a scalar multiple of that element. We list all multiplication tables of the algebras up to isomorphism.