Simple right alternative superalgebras of Abelian type whose even part is a field

We study central simple unital right alternative superalgebras $B=\Gamma\oplus M$ of Abelian type of arbitrary dimension whose even part $\Gamma$ is a field. We prove that every such superalgebra $B=\Gamma\oplus M$, except for the superalgebra $B_{1|2}$, is a double, that is, the odd part can be represented in the form $M=\Gamma x$ for a suitable $x$.
If the generating element $x$ commutes with the even part $\Gamma$, then $B$ is isomorphic to a twisted superalgebra of vector type $B(\Gamma,D,\gamma)$ introduced by Shestakov [1], [2]. But if $x$ commutes with the odd part $M$, then $B$ is isomorphic to a superalgebra $B(\Gamma, {}^*,R_\omega)$ introduced in [3] and called an $\omega$-double.
We prove that if the ground field is algebraically closed, then $B$ is isomorphic to one of the superalgebras $B_{1|2}$, $B(\Gamma,D,\gamma)$, $B(\Gamma,{}^*,R_\omega)$.