u_{ik}u_{jl}\otimes x_kx_l\\
\alpha'(x_jx_i)&=\sum_{k,l:{\varepsilon}_{kl}=1} u_{jk}u_{il}\otimes x_kx_l+\sum_{k,l:{\varepsilon}_{kl}=0} u_{jk}u_{il}\otimes x_kx_l\\
\beta'(x_kx_l)&=\sum_{i,j:{\varepsilon}_{ij}=1} u_{ik}u_{jl}\otimes x_ix_j+\sum_{i,j:{\varepsilon}_{ij}=0} u_{ik}u_{jl}\otimes x_ix_j\\
\beta'(x_lx_k)&=\sum_{i,j:{\varepsilon}_{ij}=1} u_{il}u_{jk}\otimes x_ix_j+\sum_{i,j:{\varepsilon}_{ij}=0} u_{il}u_{jk}\otimes x_ix_j\end{aligned}$$ If ${\varepsilon}_{ij}=1$, the terms $\alpha'(x_ix_j)$ and $\alpha'(x_jx_i)$ coincide. We apply $\tau_{kl}$ for $k<l$ and ${\varepsilon}_{kl}=0$ to the equation $\alpha'(x_ix_j)=\alpha'(x_jx_i)$ and we obtain (where now $x_1,x_2\in C(S^1_{{ {\mathbb R} },+})$): $$\begin{aligned}
&u_{ik}u_{jl}\otimes x_1x_2+u_{il}u_{jk}\otimes x_2x_1+u_{ik}u_{jk}\otimes x_1^2+u_{il}u_{jl}\otimes x_2^2\\
=&u_{jk}u_{il}\otimes x_1x_2+u_{jl}u_{ik}\otimes x_2x_1+u_{jk}u_{ik}\otimes x_1^2+u_{jl}u_{il}\otimes x_2^2\end{aligned}$$ By applying the maps $\eta_1$ and $\eta_2$, we obtain $u_{ik}u_{jk}=u_{jk}u_{ik}$ and $u_{il}u_{jl}=u_{jl}u_{il}$. By Lemma \[RepSphere\](a) we know $x_1x_2\neq x_2x_1$, so we finally obtain the following relations (including the case $k=l$): $${\varepsilon}_{ij}=1,{\varepsilon}_{kl}=0: \qquad\qquad u_{ik}u_{jl}=u_{jk}u_{il}$$ A similar argument using $\beta'$ yields: $${\varepsilon}_{ij}=0,{\varepsilon}_{kl}=1: \qquad\qquad u_{ik}u_{jl}=u_{il}u_{jk}$$ For ${\varepsilon}_{kl}=1$, we have $x_kx_l=x_lx_k$, hence applying $\sigma_{kl}$ to the equation $\alpha'(x_ix_j)=\alpha'(x_jx_i)$ yields the relations: $${\varepsilon}_{ij}=1,{\varepsilon}_{kl}=1: \qquad\qquad u_{ik}u_{jl}+u_{il}u_{jk}=u_{jk}u_{il}+u_{jl}u_{ik}$$ Applying it on $\beta'(x_kx_l)=\beta'(x_lx_k)$ yields the relations: $${\varepsilon}_{ij}=1,{\varepsilon}_{kl}=1: \qquad\qquad u_{ik}u_{jl}+u_{jk}u_{il}= u_{il}u_{jk}+ u_{jl}u_{ik}$$ Combining these two relations, we obtain: $${\varepsilon}_{ij}=1,{\varepsilon}_{kl}=1: \qquad\qquad u_{ik}u_{jl}= u_{jl}u_{ik}$$

*Step 5: Maximality; $u$ is orthogonal in $C(G)$.*

Using the relation $\sum_k x_k^2=1$ in $C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})$, we infer: $$\begin{aligned}
1\otimes 1
&=\sum_k\alpha'(x_k^2)\\
&=\sum_{kij} u_{ki}u_{kj}\otimes x_ix_j\\
&=\sum_{ij:i\neq j}\left(\sum_k u_{ki}u_{kj}\right)\otimes x_ix_j+
\sum_{i}\left(\sum_k u_{ki}^2\right)\otimes x_i^2\end{aligned}$$ Applying $\eta_i$ to this equation, we obtain $$\sum_k u_{ki}^2=1\qquad\forall i$$ and therefore: $$\sum_{ij:i\neq j}\left(\sum_k u_{ki}u_{kj}\right)\otimes x_ix_j=0$$ If now ${\varepsilon}_{ij}=0$, applying $\tau_{ij}$ and using $x_1x_2\neq x_2x_1$ in $C(S_{{ 