},{\varepsilon}}^{n-1})\to C(G)\otimes C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})\\
&\alpha'(x_i)= \sum_k  u_{ik}\otimes x_k, \qquad\beta'(x_i)= \sum_k  u_{ki}\otimes x_k\end{aligned}$$ For proving that there is a $^*$-homomorphism $C(O_n^{\varepsilon})\to C(G)$ sending generators to generators, we will make use of the following $^*$-homomorphisms. They arise from tensor products of the identity map $\operatorname{id}:C(G)\to C(G)$ with $^*$-homomorphisms from $C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})$ to ${{\mathbb C}}$ or to $C(S_{{ {\mathbb R} },+}^1)$ respectively; we use the universal property of $C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})$ for the existence of the latter ones. We have: $$\eta_k:C(G)\otimes C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})\to C(G), 
\qquad z\otimes x_i\mapsto \begin{cases} z &\textnormal{ if }i=k\\ 0 &\textnormal{ otherwise}\end{cases}$$ Moreover, we have for ${\varepsilon}_{kl}=1$: $$\sigma_{kl}:C(G)\otimes C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})\to C(G), 
\qquad z\otimes x_i\mapsto \begin{cases} \frac{1}{\sqrt 2}z &\textnormal{ if }i=k\textnormal{ or }i=l\\
0 &\textnormal{ otherwise}\end{cases}$$ And for ${\varepsilon}_{kl}=0$ with $k<l$: $$\tau_{kl}:C(G)\otimes C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})\to C(G)\otimes C(S_{{ {\mathbb R} },+}^1), 
\quad z\otimes x_i\mapsto \begin{cases} 
z\otimes x_1 &\textnormal{ if }i=k\\
z\otimes x_2 &\textnormal{ if }i=l\\
0 &\textnormal{ otherwise}\end{cases}$$

*Step 3: Maximality; $u_{ij}=u_{ij}^*$ holds in $C(G)$.*

We observe that all generators of $C(G)$ are self-adjoint, by applying $\eta_k$ to the following equation: $$\sum_k  u_{ik}\otimes x_k=\alpha'(x_i)=\alpha'(x_i)^*=\sum_k  u_{ik}^*\otimes x_k$$

*Step 4: Maximality; the relations $R^{\varepsilon}$ hold in $C(G)$.*

Let us compute: $$\begin{aligned}
\alpha'(x_ix_j)&=\sum_{k,l:{\varepsilon}_{kl}=1} u_{ik}u_{jl}\otimes x_kx_l+\sum_{k,l:{\varepsilon}_{kl}=0} 