In order to introduce the $q$-analog version of $\kappa (\tau)$-quantized inhomogeneous orthogonal Hopf algebra from the previous section we start with reminding the orthogonal $D=1+(D-1)$ decomposition which relays on suitable change[^6] of generators in $U(\operatorname{iso}(g))$, provided that $\tau^{2}\neq0$.

In the corresponding Lie algebra basis $\{P_{\tau},P_{i},M_{\tau i},M_{ij}\}$ the algebraic relations read as $$\begin{gathered}
[M_{ij},M_{kl}]=i(g_{il}M_{jk}-g_{jl}M_{ik}+g_{jk}M_{il}-g_{ik}M_{jl}),\label{orthDec}
\\
[M_{\tau j},M_{kl}]=i(g_{jk}M_{\tau l}-g_{jl}M_{\tau k}),
\qquad
[M_{\tau j},M_{\tau l}]=0,
\\
[P_{\tau},P_{k}]=[P_{i},P_{j}]=[P_{\tau},P_{\tau}]=0,
\\
[M_{ij},P_{k}]=i(g_{jk}P_{i}-g_{ik}P_{j}),
\qquad
[M_{ij},P_{\tau}]=0,
\\
[M_{\tau j},P_{k}]=ig_{jk}P_{\tau},
\qquad
[M_{\tau j},P_{\tau}]=-i\tau^{2}P_{j},\end{gathered}$$ while the coproducts take the form (notice that now $\tau^\mu=(1,0,\dots,0)$) $$\begin{gathered}
\Delta_{\tau} (P_{\tau} )=P_{\tau}\otimes
\left(\frac{1}{\kappa}P_{\tau}+\sqrt{1+\frac{\tau^{2}}{\kappa^{2}}C}\right)
+\left(\frac{\sqrt{1+\frac{\tau^{2}}{\kappa^{2}}C}-\frac{1}{\kappa}
P_{\tau}}{1+\frac{\tau^{2}}{\kappa^{2}}P^mP_m}\right) \otimes P_{\tau}
\nonumber
\\
\phantom{\Delta_{\tau} (P_{\tau} )=}
 -\frac{\tau^{2}}{\kappa}\left(\frac{\sqrt{1+\frac{\tau^{2}}{\kappa^{2}}C}-\frac{1}{\kappa}P_{\tau}}{1+\frac{\tau^2}{\kappa^{2}}
P^mP_m}\right)P^{j} \otimes P_{j},
\label{DPtau}
\\
\Delta_{\tau}(P_{i})=P_{i}\otimes \left(\frac{1}{\kappa}
P_{\tau}+\sqrt{1+\frac{\tau^{2}}{\kappa^{2}}C}\right) +1\otimes P_{i},
\\
\Delta_{\tau}(M_{ij})=M_{ij}\otimes 1+1\otimes M_{ij}, \qquad
i,j=1,\ldots,D-1,
\\
\Delta_{\tau}(M_{\tau i})=M_{\tau i}\otimes 1
 + \left(\frac{\sqrt{1+\frac{\tau^{2}}{\kappa^{2}}C}-\frac{1}{\kappa}
P_{\tau}}{1+\frac{\tau^{2}}{\kappa^{2}}P^mP_m}\right)\otimes M_{\tau i}
\nonumber
\\
\phantom{\Delta_{\tau}(M_{\tau i})=}
+\frac{\tau^{2}}{\kappa}\left(\frac{\sqrt{1+\frac{\tau^{2}}{\kappa^{2}}C}
-\frac{1}{\kappa}P_{\tau}}{1+\frac{\tau^{2}}{\kappa^{2}} P^mP_m}\right)P^{j} \otimes M_{ij},
\label{DMtau}\end{gathered}$$ where $\tau^{2}=g_{00}$ and after normalization it can be reduced to $\pm 1$ and $C=P^{\mu}P_{\mu}=(\tau^2)^{-1}P^2_\tau+g^{i j}P_{i}P_{j}$, $\mu,\nu=\tau,i$. For this system of generators one has $$\begin{gathered}
\Pi_{\tau}=\frac{1}{\kappa}P_{\tau}+\sqrt{1+\frac{\tau^{2}}{\kappa^{2}}C},
\qquad
\Pi_{\tau}^{-1}=\frac{\sqrt{1+\frac{\tau^{2}}{\kappa^{2}}C}-\frac{1}{\kappa}P_{\tau}}{1+\frac{\tau^{2}}{\kappa^{2}}P^mP_m},
\label{Pi}
\\
C_{\tau}=\frac{\kappa^{2}}{\tau^{2}}\left(\Pi_{\tau}+\Pi_{\tau}^{-1}-2+\frac{\tau^{2}}{\kappa^{2}} P^mP_m
\Pi_{\tau}^{-1}\right)=\frac{2\kappa^{2}}{\tau^{2}}\left(\sqrt{1+\frac{\tau^{2}}{\kappa^{2}}C} -1\right),
\label{Casimir}\end{gathered}$$ as well as the following antipodes $$\begin{gathered}
S_{\tau}\left(P_{\tau}\right)
=-\left(P_{\tau}+\frac{\tau^{2}}{\kappa}\left(C+\frac{1}{2\kappa}P_{\tau}C_{\tau}\right)\right) \Pi_{\tau}^{-1},
\label{SPtau}
\\
S_{\tau}(P_{i})=-P_{i}\Pi_{\tau}^{-1},
\qquad
S_{\tau}(\Pi_{\tau})=\Pi_{\tau}^{-1},
\\
S_{\tau}(M_{ij})=-M_{ij},
\qquad
S_{\tau}(M_{\tau i})=-M_{\tau i}-\frac{\tau^{2}}{\kappa}P^{\alpha}M_{\alpha i}-\frac{\tau^{2}}{2\kappa^{2}}C_{\tau}M_{\tau i}.
\label{SMtau}\end{gathered}$$ The relations – constitute the same Hopf 