New examples of simple Jordan superalgebras over an arbitrary field of characteristic 0

In a joint paper with the author, I. P. Shestakov constructed a new example of a unital simple special Jordan superalgebra over the real number field. It turned out that this superalgebra is a subsuperalgebra of a Jordan superalgebra of the vector type $J(\Gamma ,D)$, but it is not isomorphic to a superalgebra of this type. Moreover, the superalgebra of quotients of the constructed superalgebra is isomorphic to a Jordan superalgebra of vector type. Later, a similar example was constructed for Jordan superalgebras over a field of characteristic 0 in which the equation $t^2+1=0$ is unsolvable. In the present paper, an example is given for a Jordan superalgebra with the same properties over an arbitrary field of characteristic 0. A similar example was discovered also for a Cheng–Kac superalgebra.