{\mathbb R} }}^{n-1}$ and that $O_n^+$ acts maximally on the noncommutative sphere $S_{{ {\mathbb R} },+}^{n-1}$, we observe that $O_n^{\varepsilon}$ acts maximally on the ${\varepsilon}$-sphere $S_{{ {\mathbb R} },{\varepsilon}}^{n-1}$.

\[Action\] The ${\varepsilon}$-orthogonal quantum group $O_n^{\varepsilon}$ acts on the ${\varepsilon}$-sphere $S_{{ {\mathbb R} },{\varepsilon}}^{n-1}$ by the natural left and right actions $$\alpha: C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})\to C(O_n^{\varepsilon})\otimes C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1}),\qquad x_i\mapsto \sum_k  u_{ik}\otimes x_k$$ and: $$\beta: C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})\to C(O_n^{\varepsilon})\otimes C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1}),\qquad x_i\mapsto \sum_k  u_{ki}\otimes x_k$$ Moreover, $O_n^{\varepsilon}$ is maximal with these actions in the sense that whenever $G$ is a compact matrix quantum group acting on $S_{{ {\mathbb R} },{\varepsilon}}^{n-1}$ in the above way, then $G\subset O_n^{\varepsilon}$.

*Step 1: Existence of $\alpha$ and $\beta$.*

We put $y_i:=\sum_k u_{ik}\otimes x_k$ and compute for ${\varepsilon}_{ij}=1$: $$\begin{aligned}
y_iy_j
&=\sum_{k,l:{\varepsilon}_{kl}=1} u_{ik}u_{jl}\otimes x_kx_l+\sum_{k,l:{\varepsilon}_{kl}=0} u_{ik}u_{jl}\otimes x_kx_l\\
&=\sum_{k,l:{\varepsilon}_{kl}=1} u_{jl}u_{ik}\otimes x_lx_k+\sum_{k,l:{\varepsilon}_{kl}=0} u_{jk}u_{il}\otimes x_kx_l\\
&=\sum_{k,l} u_{jl}u_{ik}\otimes x_lx_k\\
&=y_jy_i\end{aligned}$$ Furthermore, $y_i^*=y_i$ and $\sum_i y_i^2=1$ by an easy computation using only the relations of $O_n^+$. Thus, $\alpha$ exists by the universal property. Likewise we deduce the existence of $\beta$.

*Step 2: Maximality; definition of auxiliary maps.*

Now, let $G$ be another compact matrix quantum group acting on $S_{{ {\mathbb R} },{\varepsilon}}^{n-1}$ via: $$\begin{aligned}
&\alpha',\beta': C(S_{{ {\mathbb R} 