Restrictive, split and unital quasi-Jordan algebras

It is well known that by means of the right and left products of an associative dialgebra we can build a new product over the same vector space with respect to which it becomes a right version of a Jordan algebra (in fact, this new product is right commutative) called quasi-Jordan algebra. Recently, Kolesnikov and Bremner independently have discovered an interesting property of this new product. As the results of this paper indicate, when the said property is incorporated as an axiom in the definition of quasi-Jordan algebra then in a natural way one can introduce and study concepts in this new structure such as derivations (in particular inner derivations), quadratic representations, and the structure groups of a quasi-Jordan algebra.