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 $\mathcal{P}\mathcal{C}(\eta )$-axial algebras, where $\eta$ is an element of the ground field. As a first step towards their classification, we describe $2-$ and $3$-generated subalgebras of such algebras.