are mutually non-isomorphic irreducible $\cA$-modules and $V_\tau\cong
\Hom_{\cA}(U_\tau,U)$. Since the decomposition of $U$ into irreducibles is homogeneous, the multiplicity spaces $V_\tau$ are $\Ztwo$-graded, whereas the $U_\tau$ are purely even. By Artin-Wedderburn theory [@Jacobson], $\cA=\bigoplus_{\tau}
\mathrm{End}_\C(U_\tau)\otimes 1_{V_\tau}^{}$. Thus by Remark \[rmk-dblcomm\], $
\cA'=\bigoplus_{\tau}1_{U_\tau}^{}\otimes \mathrm{End}_\C(V_\tau)
$ and $
(\cA')'=
\bigoplus_{\tau}
\mathrm{End}_\C(U_\tau)\otimes 1_{V_\tau}^{}=\cA
$.

\[lem-supcmmt\] For every $d\geq 1$, the superalgebra $\mathrm{End}_\g g(V^{\otimes d})$ is spanned by the operators $T^\sigma_{V,d}$ defined in . In particular, $
\mathrm{End}_\g g(V^{\otimes d})
=
\mathrm{End}_\g g(V^{\otimes d})_\eev
=\Mor_{\g g\text{-\upshape mod}}(V^{\otimes d},
V^{\otimes d}
)
$.

Set $\cA:=\mathrm{Span}_\C\{T_{V,d}^\sigma\ :\ \sigma\in S_d\}\subset\mathrm{End}_\C(V^{\otimes d})_\eev$. Since $\cA$ is a homomorphic image of the group algebra $\C[S_d]$, it is a semisimple associative algebra, and Lemma \[UAcomm\] implies that $(\cA')'=\cA$.

Let $\cB$ be the associative subalgebra of $\mathrm{End}_\C(V^{\otimes d})$ generated by the image of $\bfU(\g g)$. From [@sergeev Theorem 1] or [@BereleRegev Theorem 4.14] it follows that $\cA'=\cB$. Therefore we obtain that $\mathrm{End}_\g g(V^{\otimes d})=\cB'=(\cA')'=\cA$.

The canonical isomorphism $$\label{V2dV*2dE}
V^{\otimes 2d}\otimes V^{*\otimes 2d}\cong \mathrm{End}_\C(V^{\otimes 2d})$$ given in maps $v_1\otimes\cdots\otimes v_{2d}\otimes
v_1^*\otimes \cdots\otimes v_{2d}^*\in V^{\otimes 2d}\otimes V^{*\otimes 2d} 
$ to the linear map $$V^{\otimes 2d} \to V^{\otimes 2d}\ ,\ w_1\otimes \cdots\otimes w_{2d}
\mapsto \lag v^*_{2d},w_1\rag
\cdots \lag v_1^*,w_{2d}\rag
v_1\otimes\cdots\otimes v_{2d} .$$ For every $\sigma\in 