$\mathfrak{J}_{ij}^{\;\;.2} \cdot e_i$ is an ideal of $\mathfrak{J}_{ii}$

Question

Let $\mathfrak{J}$ be a Jordan algebra, and $\mathfrak{J} = \sum \mathfrak{J}_{ij}$ the Peirce decomposition of $\mathfrak{J}$ relative to orthogonal idempotents $e_i$ with sum $1$. Prove that $\mathfrak{J}_{ij}^{\;\;.2} \cdot e_i$ is an ideal of $\mathfrak{J}_{ii}$.

I proved $\mathfrak{J}_{ij}^{\;\;.2} \cdot e_i$ is contained in $\mathfrak{J}_{ii}$, and just I need to prove that $a \cdot x \in \mathfrak{J}_{ij}^{\;\;.2} \cdot e_i$, with $a = a_{ii} \in \mathfrak{J}_{ii}$ and $x=(b_{ij}c_{ij})e_i \in \mathfrak{J}_{ij}^{\;\;.2}$. In other words, $a_{ii}[(b_{ij}c_{ij})e_i] \in \mathfrak{J}_{ij}^{\;\;.2} \cdot e_i$.

Thanks.

Answer 1

In Jacobson book, Structures and Representations of Jordan Algebras, has 13 properties on page 121.