Jordanian double extensions of a quadratic vector space and symmetric Novikov algebras

First, we study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra. We give an isomorphic characterization of 2-step nilpotent pseudo-Euclidean Jordan algebras. Next, we find a Jordan-admissible condition for a Novikov algebra $\N$. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7.