introducing new generators $\tilde x^\alpha=(A^{-1})^\alpha_\mu x^\mu$ and new components $\tilde
\tau^\alpha=(A^{-1})^\alpha_\mu \tau^\mu$ ($\tilde e_\alpha=A_\alpha^\mu e_\mu$, $A^\mu_\alpha\in {\rm GL}(D, \mathbb{R})$) one preserves the form of  as well as the reality condition. It shows that the real algebra  is, in fact, independent of the components of the vector $\tau\neq0$ (for $\tau=0$ one obtains undeformed Abelian algebra). In particular, one can always reach the well-known standard form of the $\kappa$-Minkowski spacetime algebra[^13] $$\begin{gathered}
\left[x^{0}, x^{i}\right]=\frac{i}{\kappa} x^{i},
\qquad
\left[x^{i}, x^{j}\right]=0,
\qquad
i,j=1,\ldots,D-1.
\label{kM3}\end{gathered}$$ One can conclude that up to the isomorphism mentioned above, for any dimension there is only one real $\kappa$-Minkowski spacetime algebra $\mathcal{M}^D_{\kappa,\tau}$, which is covariant as a Hopf module algebra with respect to the action  of different, in general, Hopf algebras equipped with different reality structures. In fact, to be more precise, we have to distinguish three non-isomorphic options (incarnations): each one is adopted to the corresponding form of quantum $U(\operatorname{iso}(g))$:

i\) $\mathcal{M}^D_{\kappa,\tau}$ is closed in $h$-adic topology (see [@BPsigma]);

ii\) $\mathcal{M}^D_{q,\tau}$ is considered as an algebra over polynomial ring $\mathbb{C}[{1\over\kappa}]$ admitting only polynomial expressions in the formal variable ${\frac{1}{\kappa}}$ ($q$-analog version);

iii\) we assign to $\kappa$ numerical value. In this case a value of $\kappa$ becomes irrelevant since it can 