They generate the undeformed Lorentz (if extended by $p_{\mu}$, Poincaré) algebra, different from the case (i). Eq. or , together with $\mathcal{R}$, gives $$\Delta \widetilde{M}_{i0}= x_ip_0\otimes Z^{\lambda} +Z^{-(1-\lambda)}\otimes x_ip_0 -x_0p_i\otimes Z^{-\lambda} -Z^{1-\lambda}\otimes x_0p_i -(1-\lambda)a_0 p_i\otimes x_k p_kZ^{-\lambda} +\lambda a_0 x_k p_k Z^{1-\lambda}\otimes p_i.$$ An illustration of the method is given by calculating the coproduct of $M_{ij}$ (which is the same in all three cases): $$\begin{aligned}
\Delta M_{ij} &=& \Delta x_i\Delta p_j- \Delta x_j\Delta p_i \nonumber \\
&=& x_ip_j\otimes 1+ \underbrace{x_iZ^{1-\lambda}\otimes p_jZ^{\lambda}}_{1\otimes x_ip_j} -x_jp_i\otimes 1 -\underbrace{x_jZ^{1-\lambda}\otimes p_iZ^{\lambda}}_{1\otimes x_jp_i} \nonumber \\
&=& M_{ij}\otimes 1+1\otimes M_{ij}.\end{aligned}$$ The same result is trivially obtained using the twist because $\Delta_0M_{ij}$ is primitive and $M_{ij}$ commutes with $A$ and $x_kp_k$.

Expressions for the coproducts obtained in (i), (ii) and (iii) coincide with the known results in the literature. The cases (i) and (ii) are the examples of $\kappa$-Poincaré Hopf algebra, while in the case (iii) the algebra needs to be extended to $\mathfrak {igl}(4)$ in order to get a $\kappa$-deformed $\mathfrak{igl}$(4) Hopf algebra, consistent with $\kappa$-Minkowski spacetime.

Flip operator and the universal R-matrix
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The ordinary flip operator $\tau_0 :\ H \otimes H \longrightarrow H \otimes H$ is defined by $\tau_0 (h_1\otimes h_2)=h_2\otimes h_1,$ for any two 