On subalgebras of the Griess algebra with alternating Miyamoto group

We use Majorana representations to study the subalgebras of the Griess algebra that have shape (2B, 3A, 5A) and whose associated Miyamoto groups are isomorphic to An. We prove that these subalgebras exist only if n ∈ {5, 6, 8}. The case n = 5 was already treated by Ivanov, Seress, McInroy, and Shpectorov. In case n = 6 we prove that these algebras are all isomorphic and provide their precise description. In case n = 8 we prove that these algebras do not arise from standard Majorana representations.