of the twist, Eq.(\[twist\]), while the second method uses the realizations (\[realizationmi0\]) and (\[realizationmij\]) and the homomorphism of the coproduct $\Delta$, $$\Delta \hat{M}_{i0} = \Delta x_{i}\Delta p_{0}\Delta F_{1} -\Delta x_{0}\Delta p_{i}\Delta F_{2} +a_{0}\Delta x_{k}\Delta p_{k}\Delta p_{i}\Delta F_{3} +a_{0}\Delta x_{i}\Delta \vec{p}^{2}\Delta F_{4}. \label{dm2}$$ In Eq.(\[dm1\]) $\Delta_{0}\hat{M}_{i0}$ is not simply $\hat{M}_{i0}\otimes
1+1\otimes \hat{M}_{i0}$, but needs to be calculated by using Eqs. and ($\Delta_0 \hat{M}_{i0} =\Delta_0x_i \Delta_0p_0 \Delta_0F_1 -...$), and $\Delta x, \, \Delta p$ are given by Eqs.-. The coproduct of $M_{ij}$, calculated in an analogous way, is primitive.

We consider three examples of the above\
(i) $F_1=\frac{Z^{2-\lambda}-Z^{-\lambda}}{2A},\, F_2=Z^{\lambda} ,\,F_3=(1-\lambda)Z^{\lambda},\, F_4=-\frac{Z^{\lambda}}{2}$.\
This is the case of the undeformed Lorentz algebra which, if extended by noncommutative coordinates, forms a Lie algebra [@Kovacevic:2012an]. For this case we use $M$ instead of $\hat{M}$. Using Eq. or and the relations $\mathcal{R}$, gives $$\Delta M_{i0}=M_{i0}\otimes1+Z\otimes M_{i0}-a_0Z^{\lambda} p_j\otimes M_{ij}.$$ (ii) $F_1=\frac{\sinh A}{A},\, F_2=1,\, F_3=F_4=0, \, \lambda=\frac{1}{2}$\
In this example (known as the standard basis in Refs.[@Luk1]) the Lorentz algebra is deformed ($[\hat{M}_{i0},\hat{M}_{j0}] =-iM_{ij}\cosh
A$). $\Delta \hat{M}_{i0}$, calculated by or and $\mathcal{R}$ (relations and for $\lambda=\frac{1}{2}$), is $$\Delta\hat{M}_{i0}= \hat{M}_{i0}\otimes Z^{-\frac12}+ Z^\frac12\otimes\hat{M}_{i0} +\frac{a_0}2\left(M_{ij}Z^\frac12\otimes p_j- p_j\otimes M_{ij}Z^{-\frac12}\right).$$ (iii) $F_1=F_2=1, \, F_3=F_4=0$\
We denote the generators appropriate to this case by $\widetilde{M}$. 