Primitive axial algebras of Jordan type

An axial algebra over the field $\mathbb{F}$ is a commutative algebra generated by idempotents whose adjoint action has multiplicity-free minimal polynomial. For semisimple associative algebras this leads to sums of copies of $\mathbb{F}$. Here we consider the first nonassociative case, where adjoint minimal polynomials divide $(x-1)x(x-\eta )$ for fixed $0\mathrm{\ne }\eta \mathrm{\ne }1$. Jordan algebras arise when $\eta =\frac{1}{2}$, but our motivating examples are certain Griess algebras of vertex operator algebras and the related Majorana algebras. We study a class of algebras, including these, for which axial automorphisms like those defined by Miyamoto exist, and there classify the $2$-generated examples. For $\eta \mathrm{\ne }\frac{1}{2}$ this implies that the Miyamoto involutions are $3$-transpositions, leading to a classification.