the coproduct. Under the action the algebra  becomes a covariant quantum space ($\equiv$ Hopf module algebra) in a sense of the compatibility condition (a.k.a.generalized Leibniz rule) $$\begin{gathered}
L\triangleright ({x}\cdot{y})=(L_{(1)}\triangleright {x})\cdot(L_{(2)}\triangleright {y})
\label{kM2}\end{gathered}$$ as well as $$\begin{gathered}
(L\cdot M) \triangleright {x}=L\triangleright \left(M\triangleright {x}\right)
\label{kM22}\end{gathered}$$ for any ${x}, {y}\in \mathcal{M}_{\kappa,\tau}^{D}$, $L,M\in U_{\kappa,\tau}(\operatorname{iso}(g))$, where, for simplicity, we have used Sweedler type notation for the coproduct: $\Delta_{\tau}(L)=L_{(1)}\otimes L_{(2)}$. On the other hand, the presentation of the algebra  provides its natural real form by the requirement that the generators $({x}^{\mu})^{\dagger}={x}^{\mu}$ are self-adjoint in analogy to . The compatibility condition between two real structures can be expressed (see e.g. [@Sitarz_star]) as a reality condition $$\begin{gathered}
L^{\ast}\triangleright {x}^{\dagger}=\big(S_{\tau}(L^{\ast})^{\ast}\triangleright {x}\big)^{\dagger}
\label{RC3}\end{gathered}$$ for the corresponding representation (module structure). This property is enough to be checked on Hermitean generators: $L\in (M_{\mu \nu},P_{\rho})$, ${x}\in ({x}^{\mu})$ since formula  is consistent with both multiplications, i.e.  and .

It should be observed that the metric components are not involved in the definition of , so the algebra is independent of the metric itself and the metric signature in particular. One can see it also by making use of a general covariance in order to change the system of generators in (\[kMtau\]) (cf. ). Indeed, 