  A Jordan algebra J is said to be pseudo-euclidean if J is endowed with an
associative non-degenerate symmetric bilinear form B. B is said an associative
scalar product on J. First, we provide a description of the pseudo-euclidean
Jordan K-algebras in terms of double extensions and generalized double
extensions. In particular, we shall use this description to construct all
pseudo-euclidean Jordan algebras of dimension less than or equal to 5. And
then, from one of these algebras, we shall construct a twelve dimension Lie
algebra by the "TKK" construction. Second, a description of symplectic
pseudo-euclidean Jordan algebras is provided and finally we describe a
particular class of these algebras namely the class of symplectic Jordan-Manin
Algebras. In addition to these descriptions, this paper demonstrates that these
last two classes are identical and provides several information on the
structure of pseudo-euclidean Jordan algebras.
