{\mathbb R} },+}^1)$ yields: $$\sum_k u_{ki}u_{kj}=0$$ If ${\varepsilon}_{ij}=1$, then $x_ix_j=x_jx_i$ and using $\sigma_{ij}$ we deduce: $$\sum_k u_{ki}u_{kj}+\sum_k u_{kj}u_{ki}=0$$ But as $u_{ki}u_{kj}=u_{kj}u_{ki}$, we infer: $$\sum_k u_{ki}u_{kj}=0$$ Performing similar computations for $\beta'$, this proves orthogonality of $u$ and we may conclude that there is a $^*$-homomorphism $C(G)\to C(O_n^{\varepsilon})$ sending generators to generators; hence $G\subset O_n^{\varepsilon}$.

The ${\varepsilon}$-symmetric quantum group $S_n^{\varepsilon}$
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Having defined an ${\varepsilon}$-version of the orthogonal group $O_n$ by quotienting out the relations $R^{\varepsilon}$ from $O_n^+$, it is natural to define ${\varepsilon}$-versions of quantum subgroups of $O_n^+$ in the same way. In order to do so for the symmetric (quantum) group, we observe that several natural relations are equivalent, as will be discussed in the sequel.

The relations $\mathring R^{\varepsilon}$
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Recall the relations $R^{\varepsilon}$ from Definition \[DefEspOn\]: $$u_{ik}u_{jl}=\begin{cases}
u_{jl}u_{ik}&\text{if  ${\varepsilon}_{ij}=1$ and ${\varepsilon}_{kl}=1$}\\
u_{jk}u_{il}&\text{if  ${\varepsilon}_{ij}=1$ and ${\varepsilon}_{kl}=0$}\\
u_{il}u_{jk}&\text{if  ${\varepsilon}_{ij}=0$ and ${\varepsilon}_{kl}=1$}
\end{cases}$$

We now define some simpler relations.

\[DefRRing\] We define the relations $\mathring R^{\varepsilon}$ by: $$u_{ik}u_{jl}=\begin{cases}
u_{jl}u_{ik}&\text{if  ${\varepsilon}_{ij}=1$ and ${\varepsilon}_{kl}=1$}\\
0&\text{if  ${\varepsilon}_{ij}=1$ and ${\varepsilon}_{kl}=0$}\\
0&\text{if  ${\varepsilon}_{ij}=0$ and ${\varepsilon}_{kl}=1$}\end{cases}$$ In fact, we may also express them as: $$\begin{aligned}
&(\mathring R^{\varepsilon}1) \quad u_{ik}u_{jl}=u_{jl}u_{ik}\textnormal{ if } {\varepsilon}_{ij}=1 \textnormal{ and } {\varepsilon}_{kl}=1\\
&(\mathring R^{\varepsilon}2) \quad \delta_{{\varepsilon}_{kl}=0}u_{ik}u_{jl}=\delta_{{\varepsilon}_{ij}=0}u_{ik}u_{jl}\end{aligned}$$

\[LemREpsAeqRRing\] For any quantum subgroup $G_n\subset O_n^+$, 