Quadro-quadric Cremona maps and varieties 3-connected by cubics: semi-simple part and radical

Via the XJC correspondence proved in [21] we provide some structure theorems for quadro-quadric Cremona transformations and for extremal varieties 3-covered by twisted cubics by reinterpreting for these objects the algebraic results on the solvability of the radical of Jordan algebras. In this way we can define the semi-simple part and the radical part of a quadro-quadric Cremona transformation, respectively of an extremal variety 3-covered by twisted cubics, and then describe how general objects are constructed from the semi simple ones, which are completely classified modulo certain equivalences, via suitable null radical extensions.