A symmetric biderivation structure on polynomial algebras and a class of modules over the special Jordan algebra $H_n(K)$ of symmetric matrices

There exists a biderivation structure on the polynomial algebra $\mathcal{A}[n]=K[x_1,\dots ,x_n],$ where $K$ is a field with $\mathrm{char}(K)\mathrm{\ne }2$, defined by $f\circ h={\sum }_{i=1}^n\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_i}\frac{\mathrm{\partial }h}{\mathrm{\partial }{x}_i}\mathrm{.}$ Let ${\mathcal{A}}_k[n]$ denote the subspace of homogeneous polynomials of degree $k$. Then $({\mathcal{A}}_2[n],\circ )$ is a Jordan algebra, isomorphic to the special Jordan algebra $H_n(K)$ of $n\times n$ symmetric matrices. Each ${\mathcal{A}}_k[n]$ is a natural ${\mathcal{A}}_2[n]$-bimodule, which admits a weight space decomposition with respect to a complete set of mutually orthogonal idempotents. In particular, the weight space decomposition of ${\mathcal{A}}_2[n]$ coincides with its Peirce decomposition. ${\mathcal{A}}_k[n]$ is a Jordan bimodule if and only if $k=0,1,2$. Equivalently, for all $k\ge 3$, ${\mathcal{A}}_k[n]$ is not a Jordan bimodule. The group of algebra automorphisms of $(\mathcal{A}[n],\cdot ,\circ )$ that preserve each homogeneous component ${\mathcal{A}}_k[n]$ is isomorphic to the orthogonal group $O(n,K)$. If $\mathrm{char}(K)=0$, then the algebra $(\mathcal{A}[n],\cdot ,\circ )$ is simple, i.e., it has no nonzero proper ideals. Moreover, in this case, each ${\mathcal{A}}_k[n]$ is a simple ${\mathcal{A}}_2[n]$-bimodule.