  "Fusion rules" are laws of multiplication among eigenspaces of an idempotent.
This terminology is relatively new and is closely related to primitive axial
algebras, introduced recently by Hall, Rehren, and Shpectorov. Axial algebras,
in turn, are closely related to $3$-transposition groups and vertex operator
algebras.
  In earlier work we studied primitive axial algebras, not necessarily
commutative, and showed that they all have Jordan type. In this paper, we show
that all finitely generated primitive axial algebras are direct sums of
specifically described flexible finite dimensional noncommutative algebras, and
commutative axial algebras generated by primitive axes of the same type. In
particular,all primitive axial algebras are flexible. They also have Frobenius
forms. We give a precise description of all the primitive axes of axial
algebras generated by two primitive axes.
