Universal Axial Algebras and a Theorem of Sakuma

In the first half of this paper, we define axial algebras: nonassociative commutative algebras generated by axes, that is, semisimple idempotents---the prototypical example of which is Griess' algebra [C85] for the Monster group. When multiplication of eigenspaces of axes is controlled by fusion rules, the structure of the axial algebra is determined to a large degree. We give a construction of the universal Frobenius axial algebra on $n$ generators with a specified fusion rules, of which all $n$-generated Frobenius axial algebras with the same fusion rules are quotients. In the second half, we realise this construction in the Majorana / Ising / $\mathrm{Vir}(4,3)$-case on $2$ generators, and deduce a result generalising Sakuma's theorem in VOAs [S07].