  Axial algebras are non-associative algebras generated by semisimple
idempotents whose adjoint actions obey a fusion law. Axial algebras that are
generated by two such idempotents play a crucial role in the theory. We
classify all primitive 2-generated axial algebras whose fusion laws have two
eigenvalues and all graded primitive 2-generated axial algebras whose fusion
laws have three eigenvalues. This represents a significant broadening in our
understanding of axial algebras.
