$\tau^2\neq0$. First of all one can prove that different values of $\kappa$ give rise to the same (isomorphic) Hopf algebras. It can be seen if one changes the generators by the re-scaling $P_i\mapsto {\frac{1}{\kappa}}P_i$ which is equivalent to setting $\kappa=1$[^9].

One can argue that this new Hopf algebra has some advantages with respect to the previous ones. For example, in the (semi-)simple case there is a duality between Drinfel’d–Jimbo quantized enveloping algebras and some matrix (or coordinate) Hopf algebras. The same is expected for the case above [@PW-2; @PW-1].

Moreover, one can also introduce an element $P_{\tau}$ expressed in terms of $q$-analog algebra generators by the formula . This means that $U_{q,\tau}(\operatorname{iso}(g))$ contains as a subalgebra the universal enveloping algebra $U(\operatorname{iso}(g))$. However it is not a sub-Hopf algebra. This fact is meaningful for the representation theory of $U_{q,\tau}(\operatorname{iso}(g))$: any representation of $U_{q,\tau}(\operatorname{iso}(g))$ becomes automatically a representation of the corresponding Lie algebra $\operatorname{iso}(g)$. The inverse statement is in general not true. Instead one gets the following selection rule: a representation of an orthogonal Lie algebra $\operatorname{iso}(g)$ becomes at the same time a representation of its quantum version provided the elements $\Pi_\tau$, $\Pi_\tau^{-1}$ as given by the formula  are well defined (self-adjoint) 