2-generated axial algebras of Monster type

We provide the basic setup for the project, initiated by Felix Rehren, aiming at classifying all 2-generated axial algebras of Monster type (α,β) over a field F. Using this, we first show that every such algebra has dimension at most 8, except for the case (α, β) = (2, 12 ), where the Highwater algebra provides examples of dimension n, for all n ∈ N ∪ {∞}. We then classify all 2-generated axial algebras of Monster type (α,β) over Q(α,β), for α and β algebraically independent over Q. Finally, we generalise the Norton-Sakuma Theorem to every primitive 2-generated axial algebra of Monster type (1/4, 1/32 ) over a field of characteristic zero, dropping the hypothesis on the existence of a Frobenius form.