\hat{x}_\mu =\hat{x}_\mu \otimes 1= Z^{-1}\otimes \hat{x}_\mu -a_\mu p_\alpha^{L}\otimes \hat{x}^{\alpha},$$ where $p_0^{L}=(1-Z^{-1})/a_0$ and $p_i^{L}=p_iZ^{\lambda-1}$. The calculated coproducts are in accordance with the Leibnitz rule for $\hat{x}_\mu$ (see Eq.(31) in [@km11]).

Generalized Poincaré algebras and coalgebras {#IV}
============================================

There are infinitely many ways of implementing (deformed) Poincaré algebras compatible with the $\kappa$-Minkowski spacetime [@Kovacevic:2012an]. If one writes an ansatz for the Lorentz generators in the form $$\begin{aligned}
 \label{realizationmi0} 
\hat{M}_{i0} &=& x_{i}p_{0}F_{1}\left(A,b\right) -x_{0}p_{i}F_{2}\left( A,b\right) +a_{0}\left( x_{k}p_{k}\right) p_{i}F_{3}\left( A,b\right) +a_{0}x_{i}\vec{p}^{2}F_{4}\left( A,b\right) \\
\hat{M}_{ij} &=& M_{ij}=x_{i}p_{j}-x_{j}p_{i}, \label{realizationmij} \end{aligned}$$ the (deformed) commutation relations can be found in [@Kovacevic:2012an]. The form of the Lorentz generators given by (\[realizationmi0\]) is compatible with the deformed Lorentz algebra (with $\kappa$-Poincaré algebra included), where the Minkowski metric is replaced with the more general metric, having generally nonlinear dependence on momenta. This is unlike the form of the metric dependence of momenta that emerged within the idea of relative locality [@AmelinoCamelia:2011bm],[@rel_loc], where the metric dependence on momenta is linear.

Following the results of the previous section (relations $\mathcal{R}$, $\Delta_0$ and $\Delta$ for $x_{\mu}$ and $p_{\mu}$), we present two (equivalent) new methods for calculating the coproduct of $\hat{M}_{i0}$. The first method includes the twist deformation $$\Delta \hat{M}_{i0} = \mathcal{F}\Delta_{0}\hat{M}_{i0}\mathcal{F}^{-1} \label{dm1}$$ with the help 