by the same factor does not change formulas – involving these symbols, so it can be treated as an isomorphism of the corresponding Hopf algebras, i.e. $U_{\kappa,\tau}(\operatorname{iso}(g))\cong U_{\lambda\kappa,\lambda\tau}(\operatorname{iso}(g))$. As a practical application one finds that the vector $\tau$ can be normalized to the values $\tau^2=\pm 1, 0$.

This unified description has the advantage of general covariance manifested via tensorial character of all defining formulas –. Consider a change of basis in the space $V$: $e_{\mu}\mapsto \tilde{e}_{\mu}=A_{\mu}^{\alpha}e_{\alpha}$ by a non-degenerate matrix $A_{\alpha}^{\beta}\in {\rm GL}(D,\mathbb{R})$. Thus one can introduce the new generators $$\begin{gathered}
\tilde{P}_{\alpha}=A_{\alpha}^{\mu}P_{\mu},
\qquad
\tilde{M}_{\alpha \beta}=A_{\alpha}^{\mu}A_{\beta}^{\nu}M_{\mu \nu}
\label{cov}\end{gathered}$$ together with $\tilde{g}_{\alpha \beta}=A_{\alpha}^{\mu}A_{\beta}^{\nu}g_{\mu \nu}$, $\tilde{\tau}_{\alpha}=A_{\alpha}^{\mu}\tau_{\mu}$ (but $\tilde{\tau}^{\alpha}=(A^{-1})_{\mu}^{\alpha}\tau^{\mu}$, $\tilde{C}=C$ and therefore ). Then all formulas of this section remain valid if we replace all objects without tilde sign by the corresponding ones with the tilde. Moreover the real structure  is preserved. It means that $U_{\kappa,\tau}(\operatorname{iso}(g))\cong
U_{\kappa,\tilde{\tau}}(\operatorname{iso}(\tilde{g}))$ as real Hopf algebras. In particular, if $A_{\alpha}^{\beta}\in O(g)$ then (internal automorphism). This fact is important for possible physical applications and interpretations (see last section in [@BP_unif]). It is to be observed that the transformation  does not change the metric signature.

Specialization of the $\boldsymbol{\kappa(\tau)}$-inhomogeneous orthogonal\
Hopf algebras
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It is known that quantized enveloping Lie algebras 