On new proper Jordan schemes related to quaternion and octonion algebras

Abstract: We present a construction of a Jordan scheme from an elementary abelian $2$-group of rank $n$ and a ${1,-1}$-matrix of order $2^n$ that satisfies a specified condition. We then prove that the orders of matrices with the specified condition are limited to $2, 4$ or $8$. Using these matrices, we construct essentially two new proper Jordan schemes of orders $16$ and $32$. Finally, we analyze the structures of the real adjacency Jordan algebras of these Jordan schemes and prove that they are the first known examples whose real adjacency Jordan algebras admit simple components of type $\mathbb{R} \oplus_f \mathbb{R}^n$, namely non-Hermitian type.