A polynomial criterion for Jordan elements in a free associative algebra

Let $A=\mathrm{\Phi }⟨X⟩$ be a free associative algebra over a field of characteristic zero, and let $J$ be the Jordan subalgebra of $A^{(+)}$ generated by $X$ and $1$. For every $n\ge 1$ we construct an element $U_n\in \mathbb{Q}[{\mathfrak{S}}_n]$ whose image on $A_n$ is exactly $J_n$. If $$\det (tI-U_n{\mathrm{\mid }}_{V_n})=t^{e_n}{q}_n(t),q_n(0)\mathrm{\ne }0,$$ on the multilinear component $V_n$, then $$a\in J_n⟺a{q}_n(U_n)=0.$$ Thus the criterion gives a finite algorithm for recognizing Jordan elements: in each degree one constructs $U_n$ and $q_n$ and tests the single equation $a{q}_n(U_n)=0$. Moreover, $I-q_n(U_n)\mathrm{/}{q}_n(0)$ is a projection of $A_n$ onto $J_n$. We give the projection explicitly in degrees at most four, record the multilinear dimensions through degree eight, and formulate the analogous criterion on each fixed multihomogeneous component.