  Motivated by Yabe's 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 Yabe's $\mathrm{III}(\alpha,\frac{1}{2},
\delta)$ 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 Yabe's examples,
we use our new viewpoint to identify the axet, that is, the closure of the two
generating axes.
