bar involution on ${\U^{\imath}}_q(\mathfrak{sl}_{2r+1}))$ by ${\psi_{\imath}}$.)

\[int:prop:embedding\] There is an injective $\Qq$-algebra homomorphism $\imath :  {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \rightarrow \U_q(\mathfrak{sl}_{2r+1})$ defined by, for all $i \in \I^{\imath} _{2r}$, $$\begin{aligned}
\bk_{\alpha_i} \mapsto K_{\alpha_i}K^{-1}_{\alpha_{-i}}, \quad
\be_{\alpha_i} \mapsto  E_{\alpha_i} +  F_{\alpha_{-i}}K^{-1}_{\alpha_i}, \quad
\bff_{\alpha_i} \mapsto K^{-1}_{\alpha_{-i}} F_{\alpha_i} + E_{\alpha_{-i}}.\end{aligned}$$

In [@BW13 Proposition 2.2], we consider the embedding that maps $\be_{\alpha_i}$ to $E_{\alpha_i} +  q^{-\delta_{i, \hf}} F_{\alpha_{-i}}K^{-1}_{\alpha_i}$ and maps $\bff_{\alpha_i}$ to $q^{\delta_{i, \hf}} K^{-1}_{\alpha_{-i}} F_{\alpha_i} + E_{\alpha_{-i}}$.

Note that $E_{\alpha_i} (K^{-1}_{\alpha_i}F_{\alpha_{-i}}) 
= q^{2} (K^{-1}_{\alpha_i}F_{\alpha_{-i}}) E_{\alpha_i}$ for $i \in \I^{\imath} _{2r} $. We have for $i \in \I^{\imath} _{2r} $, $$\begin{aligned}
\label{int:eq:beZ} \imath(\be^{(a)}_{\alpha_i}) 
 &= \sum^{a}_{j=0}  q^{j(a-j)}{(F_{\alpha_{-i}} K^{-1}_{\alpha_i} )^j \over [j]!}\frac{E^{a-j}_{\alpha_i}}{[a-j]!},
   \\
\label{int:eq:bffZ} \imath(\bff^{(a)}_{\alpha_i}) 
 &= \sum^{a}_{j=0}  q^{j(a-j)}{(K^{-1}_{\alpha_{-i}}F_{\alpha_{i}})^j \over [j]!}\frac{E^{a-j}_{\alpha_{-i}}}{[a-j]!}.\end{aligned}$$

\[int:prop:coproduct\] The coproduct $\Delta$ on $\U_q(\mathfrak{sl}_{2r+1})$ restricts under the embedding $\imath$ to a $\Qq$-algebra homomorphism $$\Delta : {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \longrightarrow {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \otimes \U_q(\mathfrak{sl}_{2r+1})$$ such that for all $i \in \I^{\imath} _{2r} $, $$\begin{aligned}
\Delta(\bk_{\alpha_i}) &= \bk_{\alpha_i} \otimes K_{\alpha_i} K^{-1}_{\alpha_{-i}},
 \\
\Delta({\be_{\alpha_i}}) &= 1 \otimes E_{\alpha_i} + \be_{\alpha_i} \otimes 
 K^{-1}_{\alpha_i} + \bk^{-1}_{\alpha_i} \otimes  F_{\alpha_{-i}} K^{-1}_{\alpha_i} ,
  \\
\Delta (\bff_{\alpha_i}) &= \bk_{\alpha_i} \otimes K^{-1}_{\alpha_{-i}} F_{\alpha_i}
  + \bff_{\alpha_i} \otimes K^{-1}_{\alpha_{-i}} + 1 \otimes E_{{\alpha_{-i}}}.\end{aligned}$$ Similarly, the counit $\epsilon$ of $\U_q(\mathfrak{sl}_{2r+1})$ induces a $\Qq$-algebra homomorphism $$\epsilon : {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \rightarrow \Qq$$ such that $\epsilon(\be_{\alpha_i}) =\epsilon(\bff_{\alpha_i})=0$ and 