On train algebras of rank 4

The existence of idempotent elements in train algebras of rank greater than 3 is an open question to be solved. Recent H. Guzzo results 7] on train algebras of rank 4 are based on the underlying assumption of the existence of an idempotent. In the present paper we establish the conditions that ensure the existence of such an idempotent. We also give additional properties on the Peirce decomposition which allow us to characterize some train algebras of rank 4. Finally, we give a characterization of the train algebras of rank 4 which are power-associative algebras or Jordan algebras.