Axial view on pseudo-composition algebras and train algebras of rank 3

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 [Formula: see text]-axial algebras, where [Formula: see text] 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.