formula for the coalgebra structure, in this case reduces to $$\begin{gathered}
\Delta_{\tau}\left(P\right)=P\otimes \left(1+\frac{1}{\kappa} P_{+}\right) +1\otimes P
\qquad
\text{for}
\quad
P\in \{P_{+},P_{a}\},
\label{DtauM1}
\\
\Delta_{\tau}(P_{-})=P_{-}\otimes \left(1+\frac{1}{\kappa} P_{+}\right)
+\left(1+\frac{1}{\kappa}P_{+}\right)^{-1}\otimes P_{-}
\nonumber
\\
\phantom{\Delta_{\tau}(P_{-})=}
 -\frac{1}{\kappa}\left(P_{-}+\frac{1}{2\kappa}C_{+}\right) \left(1+ \frac{1}{\kappa}P_{+}\right)^{-1}\otimes
P_{+}-\frac{1}{\kappa} P^{a}\left(1+\frac{1}{\kappa}P_{+}\right)^{-1}\otimes P_{a},
\\
\Delta_{\tau} (M_{+ -} )=M_{+ -}\otimes 1+\left(1+\frac{1}{\kappa}P_{+}\right)^{-1}\otimes
M_{+ -}-\frac{1}{\kappa}P^{a}\left(1+\frac{1}{\kappa}P_{+}\right)^{-1}\otimes M_{+ a},
\\
\Delta_{\tau} (M_{- a} )=M_{- a}\otimes 1+\left(1+\frac{1}{\kappa}P_{+}\right)^{-1}\otimes M_{- a}
\nonumber
\\
\phantom{\Delta_{\tau} (M_{- a} )=}
 -\frac{1}{\kappa}\left(P_{-}+\frac{1}{2\kappa}C_{+}\right) \left(1+ \frac{1}{\kappa}P_{+}\right)^{-1}\!\otimes
M_{+ a}-\frac{1}{\kappa} P^{b}\left(1+\frac{1}{\kappa}P_{+}\right)^{-1}\!\otimes M_{ba},\!\!\!
\\
\Delta_{\tau} (M )=M\otimes 1+1\otimes M
\qquad
\text{for}
\quad
M\in \{M_{+ a},M_{ab}\}
\label{DtauM2}\end{gathered}$$ and can be obtained by twisting of undeformed (primitive) coproducts by the extended Jordanian twist [@BP_unif] $$\begin{gathered}
\mathcal{F}=\exp  (-iM_{+ -}\otimes\ln\Pi_{+} ) \exp\left(-\frac{i}{\kappa}M_{+ a}\otimes P^{a}\Pi_{+}^{-1}\right)
\nonumber
\\
\phantom{\mathcal{F}}
=\exp \left(-\frac{i}{\kappa}M_{+ a}\otimes P^{a}\right) \exp\left(-iM_{+ -}\otimes\ln\Pi_{+}\right),
\label{twist_LC}\end{gathered}$$ which satisfies the two-cocycle condition, see also [@KLM; @Mudrov]. Moreover, the twist is unitary, i.e. the resulting deformation preserves the real form (cf. ). For this system of generators we have $\Pi_{+}\doteq 1+\frac{1}{\kappa}P_{+}$ and $C=2P_{+}P_{-}+P^{a}P_{a}$. The antipodes are as follows $$\begin{gathered}
S_{\tau} (P )=-P\Pi_{+}^{-1}
\qquad
\text{for}
\quad
P\in \{P_{+},P_{a}\},
\\
S_{\tau}(P_{-})=-P_{-}\Pi_{+}-\frac{1}{\kappa}\left(1+ \frac{1}{2\kappa}P_{+}\right)P_{a}P^{a}\Pi_{+}^{-1},
\\
S_{\tau}(M_{+-})=-\Pi_{+}M_{+-}-\frac{1}{\kappa} P^{a}M_{+a},
\\
S_{\tau} (M_{-a} )=-\Pi_{+}M_{-a}-\frac{1}{\kappa}\left(P_{-}+\frac{1}{2\kappa}C\right)M_{+a}-\frac{1}{\kappa}P^{b}M_{ba},
\\
S_{\tau} (M )=-M
\qquad
\text{for}
\quad
M\in \{M_{+a},M_{ab}\}.\end{gathered}$$ Due to the presence of $\Pi^{-1}_+$ some of the above expressions are infinite (formal) power 