Free multiplet, algebra, and the inherited symbols

The inherited symbols ${\mathcal{B}}_f=-\int f{\mathsf{X}}_1[\gamma ]\cdot \partial \varphi$ and $\mathcal{T}(f,g)[\varphi ]=\int f({\mathsf{X}}_1[r_g[\varphi ]])\cdot \partial \varphi$, $r_g=-\frac{1}{2}\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}(g{T}_0[\varphi ])$, are admitted in ${\mathcal{E}}_H$; the physical one-graviton response of ${\mathcal{B}}_f$ is $-\mathcal{T}$ at coupling degree ${\kappa }^2$ XIC. Their admission is not reopened here.