$^*$-homomorphism ${\varphi}:C(O_n^{\varepsilon})\to C(S_n)$ sending generators to generators. Composing it with $\operatorname{ev}_\sigma$ yields the following contradiction: $$1=\delta_{i\sigma(k)}\delta_{j\sigma(l)}=\operatorname{ev}_\sigma\circ{\varphi}(u_{ik}u_{jl})
=\operatorname{ev}_\sigma\circ{\varphi}(u_{jk}u_{il})=\delta_{j\sigma(k)}\delta_{i\sigma(l)}=0$$

Next, we will show that different matrices ${\varepsilon}$ give rise to different ${\varepsilon}$-orthogonal quantum groups.

\[RepOnEps\] We have the following $^*$-homomorphism: $${\varphi}_{\varepsilon}:C(O_n^{\varepsilon})\to C^*(\mathbb Z_2^{\varepsilon}),\qquad u_{ij}\mapsto \delta_{ij}z_i$$

The existence of $\pi$ is due to the universal property.

The preceding lemma is due to the fact that the diagonal subgroup of $O_n^{\varepsilon}$ is the Coxeter group $\mathbb Z_2^{\varepsilon}$. The diagonal subgroup of a compact matrix quantum group $(A,u)$ is constructed as follows. First, take the quotient of $A$ by the relations $u_{ij}=0$ for $i\neq j$. If $u$ is a unitary, so are all $u_{ii}$ in the quotient and we thus obtain the group $C^*$-algebra $C^*(G)$ of some group $G$. This group is called the diagonal subgroup of $(A,u)$.

\[OnEpsNoncommutative\] We have $O_n^{\varepsilon}\neq O_n^{{\varepsilon}'}$ for ${\varepsilon}\neq{\varepsilon}'$ in the sense that there is no $^*$-isomorphism $C(O_n^{\varepsilon})\to C(O_n^{{\varepsilon}'})$ sending generators to generators. Moreover, $C(O_n^{\varepsilon})$ is noncommutative as soon as ${\varepsilon}\neq{\varepsilon}_{\operatorname{comm}}$.

The proof is similar to the one of Theorem \[VglSphaeren\] using the maps ${\varphi}_{\varepsilon}$ of Lemma \[RepOnEps\] rather than those of Lemma \[RepSphere\](c).

Similarly to the well-known facts [@BanGos] that $O_n$ acts maximally on the commutative sphere $S_{{ 