Generalised dihedral subalgebras from the Monster

The conjugacy classes of the Monster which occur in the McKay observation correspond to the isomorphism types of certain 2-generated subalgebras of the Griess algebra. Sakuma, Ivanov and others showed that these subalgebras match the classification of vertex algebras generated by two Ising conformal vectors, or of Majorana algebras generated by two axes. In both cases, the eigenvalues parametrising the theory are fixed to $\alpha=\frac14,\beta=\frac1{32}$. We generalise these parameters and the algebras which depend on them, in particular finding the largest possible (nonassociative) axial algebras which satisfy the same key features, by working extensively with the underlying rings. The resulting algebras admit an associating symmetric bilinear form and satisfy the same 6-transposition property as the Monster; $\alpha=\frac14,\beta=\frac1{32}$ turns out to be distinguished.