Split spin factor algebras

Motivated by Yabes classification of symmetric $2$-generated axial algebras of Monster type, we introduce a large class of algebras of Monster type $(alpha, frac{1}{2})$, generalising Yabes $mathrm{III}(alpha,frac{1}{2}, delta)$ family. Our algebras bear a striking similarity with Jordan spin factor algebras with the difference being that we asymmetrically split the identity as a sum of two idempotents. We investigate the properties of this algebra, including the existence of a Frobenius form and ideals. In the $2$-generated case, where our algebra is isomorphic to one of Yabes examples, we use our new viewpoint to identify the axet, that is, the closure of the two generating axes.