On train algebras of degree 2 and exponent 4

In this paper, we deal with a class of nonassociative algebras called train algebras of degree 2 and exponent 4; Thus we give some results on algebras verifying a train identity of degree 2 and exponent 4. The structure of this class of algebras is studied through Peirce decomposition relative to a non zero idempotent. We give the necessary and sufficient conditions for an algebra verifying a train identity of degree 2 and exponent 4 to be Bernstein or train algebra of rank less than or equal to 3. Finally, we give some necessary conditions that must be verified by a train algebra of degree 2 and exponent 4, mainly in dimension four.