Weakly primitive axial algebras

In earlier work we studied the structure of primitive axial algebras of Jordan type (PAJs), not necessarily commutative, in terms of their primitive axes. In this paper we weaken primitivity and permit several pairs of (left and right) eigenvalues satisfying more general fusion rules, bringing in interesting new examples such as the band semigroup algebras and other commutative and noncommutative examples, but still strong enough to permit one to construct a nondegenerate symmetric Frobenius form. Also we broaden our investigation and describe 2-generated algebras in which only one of the generating axes is weakly primitive and satisfies our fusion rules, on condition that its zero-eigenspace is one-dimensional. We also characterize when both axes satisfy our fusion rules (weak PAJs), and describe precisely the two-dimensional axial algebras. In contrast to the previous situation, there are weak PAJs of dimension > 3 generated by two axes.