algebra although written in another presentation (system of generators). The importance of such presentation has been shown in [@BP_unif] by the relation to Majid–Ruegg formulation [@MR]. Again, above coproducts and antipodes are formal power series in $\frac{1}{\kappa}$ as well. It is to be observed that the elements $(P_\tau, \Pi_{\tau}, \Pi_{\tau}^{-1})$ are not algebraically independent, since $$\begin{gathered}
P_{\tau}=\frac{\kappa}{2}\left(\Pi_{\tau}-\Pi_{\tau}^{-1}\left(1+ \frac{\tau^{2}}{\kappa^{2}} P^mP_m\right)\right).
\label{d3}\end{gathered}$$ In the next section we show how to use this fact and by taking advantage of two (mutually inverse) group-like elements $(\Pi_\tau, \Pi_\tau^{-1})$ one can eliminate infinite power series from the formulas –.

**$\boldsymbol{q}$-analog version of $\boldsymbol{U_{\kappa,\tau}(\operatorname{iso}(g))}$: $\boldsymbol{\tau^2\neq0}$.** As a first step, inside the Hopf algebra from the previous section[^7], one can consider sub-Hopf algebra generated by elements $(M_{ij}, P_{i}, M_{\tau i}, \Pi_{\tau}$, $\Pi_{\tau}^{-1})$ and call it $U_{q,\tau}(\operatorname{iso}(g))$. Its generators satisfy the following relations[^8] $$\begin{gathered}
\Pi_{\tau}\Pi_{\tau}^{-1}=1=\Pi_{\tau}\Pi_{\tau}^{-1},
\label{q1}
\\
 [P_{i},\Pi_{\tau} ]= [M_{ij},\Pi_{\tau} ]=0,
\qquad
 [M_{\tau i},\Pi_{\tau} ]=-\frac{i}{\kappa} P_{i},
\\
 [M_{\tau i,}P_{j} ]
=ig_{ij}\frac{\kappa}{2}\left(\Pi_{\tau}-\Pi_{\tau}^{-1}\left(1+\frac{\tau^{2}}{\kappa^{2}}P^m P_{m}\right)\right).
\label{q2}\end{gathered}$$ Commutators with $\Pi_{\tau}^{-1}$ can be easily calculated from the above (e.g. $[M_{\tau i},\Pi_{\tau}^{-1}]=\frac{i}{\kappa}P_{i}\Pi_{\tau}^{-2}$).

Alternatively, one can abstractly define the algebraic structure of $U_{q,\tau}(\operatorname{iso}(g))$, in a way similar to the universal enveloping algebra, i.e. as a universal associative algebra generated by elements $(M_{ij},
P_{i}, M_{\tau i}, \Pi_{\tau}, 