  In this paper we study the structure of a class of algebras satisfying a
polynomial identity of degree 6. We show, assuming the existence of a non-zero
idempotent, that if an algebra satisfies such an identity, it admits a Peirce
decomposition related to this idempotent. We studied the algebraic structure
and highlighted the connections of the algebras of this class with Bernstein
algebras, train algebras, Jordan algebras and power associative algebras.
  Keywords: Peirce decomposition, Bernstein algebra, Jordan algebra, Power
associative algebra, train algebra,polynomial identity, idempotent.
