  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 $\mathcal{PC}(\eta)$-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.
