Connes with Dubois-Violette [@Con1] or with Landi [@Con2], also collected in the survey [@Landi]; see also [@BanCox; @BanSurv] for recent expositions about noncommutative spheres and latest references.

The extensive work of Banica on noncommutative spheres is to be highlighted, see amongst others [@BanGos; @BanCox; @BanSph; @BanPoly; @BanSurv]. However, his relations on the coordinates $x_i$ are mostly chosen in a uniform way [@BanSph Def. 2.2], [@BanCox Def. 1.7] rather than as partial relations; so our spheres appear to be new.

The ${\varepsilon}$-orthogonal quantum group $O_n^{\varepsilon}$
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The algebra of functions over the orthogonal group $O_n\subset M_n({ {\mathbb R} })$ can be viewed as the following universal $C^*$-algebra: $$C(O_n)=C^*(u_{ij}, i,j=1,\ldots,n\;|\; u_{ij}=u_{ij}^*, u \textnormal{ is orthogonal}, R^{\textnormal{comm}})$$ Here: $$\begin{aligned}
u \textnormal{ is orthogonal} &\qquad \Longleftrightarrow\qquad &\sum_k u_{ik}u_{jk}=\sum_k u_{ki}u_{kj}=\delta_{ij}\\
R^{\textnormal{comm}}&\qquad \Longleftrightarrow\qquad & u_{ij}u_{kl}=u_{kl}u_{ij} \quad \forall i,j,k,l\end{aligned}$$

Wang [@WangOrth] defined a noncommutative analogue of it, the *(free) orthogonal quantum group* $O_n^+$ given by: $$C(O_n^+)=C^*(u_{ij}, i,j=1,\ldots,n\;|\; u_{ij}=u_{ij}^*, u \textnormal{ is orthogonal})$$ For an introduction to compact matrix quantum groups, we refer to the original articles by Woronowicz [@WoCMQG; @WoRemark] or the books [@Nesh; @Tim]. In the sequel, the tensor product of $C^*$-algebras is always with respect to the minimal tensor product.

\[DefEspOn\] We define the *${\varepsilon}$-orthogonal 