We show that pseudo-composition algebras and train algebras of rank 3 generated by idempotents are characterized as axial algebras with fusion laws derived from the Peirce decompositions of idempotents in these classes of algebras. The corresponding axial algebras are called PC(η)-axial algebras, where η is an element of the ground field. As a first step toward their classification, we describe the 2- and 3-generated subalgebras of such algebras. © World Scientific Publishing Company.