\in \I^{\imath}_{2r+1}$, $a \in \N$, $$\begin{aligned}
\label{eq:beZ}
\imath(\be^{(a)}_{\alpha_i}) &= \sum^{a}_{j=0}  q^{j(a-j)}F^{(j)}_{\alpha_{-i}}K^{-j}_{\alpha_i} E^{(a-j)}_{\alpha_i},
 \\
\label{eq:bffZ}
\imath(\bff^{(a)}_{\alpha_i}) &= \sum^{a}_{j=0}  q^{j(a-j)}F^{(j)}_{\alpha_{i}}K^{-j}_{\alpha_{-i}} E^{(a-j)}_{\alpha_{-i}}.\end{aligned}$$

\[prop:coproduct\] The coproduct $\Delta$ on $\U_q(\mathfrak{sl}_{2r+2})$ restricts via the embedding $\imath$ to a $\Qq$-algebra homomorphism $$\Delta : {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \longrightarrow {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \otimes \U_q(\mathfrak{sl}_{2r+2})$$ such that, for all $i \in \I^{\imath}_{2r+1}$, $$\begin{aligned}
\Delta(\bk_{\alpha_i}) &= \bk_{\alpha_i} \otimes K_{\alpha_i} K^{-1}_{\alpha_{-i}},
 \\
\Delta({\be_{\alpha_i}}) &= 1 \otimes E_{\alpha_i} + \be_{\alpha_i} \otimes K^{-1}_{\alpha_i} 
+ \bk^{-1}_{\alpha_i} \otimes K^{-1}_{\alpha_i}F_{\alpha_{-i}},
 \\
\Delta (\bff_{\alpha_i}) &= \bk_{\alpha_i} \otimes F_{\alpha_i}K^{-1}_{\alpha_{-i}} + \bff_{\alpha_i} 
\otimes K^{-1}_{\alpha_{-i}} + 1 \otimes E_{{\alpha_{-i}}},
 \\
\Delta(\bt) &= \bt \otimes K^{-1}_{\alpha_0} + 1 \otimes q F_{\alpha_0}K^{-1}_{\alpha_0}+ 1 \otimes E_{\alpha_0}.\end{aligned}$$ Similarly, the counit $\epsilon$ of $\U_q(\mathfrak{sl}_{2r+2})$ induces a $\Qq$-algebra homomorphism $$\epsilon : {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \rightarrow \Qq$$ such that $\epsilon(\be_{\alpha_i}) =\epsilon(\bff_{\alpha_i})=0$, $\epsilon(\bt) = 0$, and $\epsilon(\bk_{\alpha_i}) =1$ for all $i \in \I^{\imath}_{2r+1}$.

The map $\Delta : {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \mapsto {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \otimes \U_q(\mathfrak{sl}_{2r+2})$ is coassociative, i.e., we have $(1 \otimes \Delta) \Delta = (\Delta \otimes 1)\Delta: {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \longrightarrow {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \otimes \U_q(\mathfrak{sl}_{2r+2}) \otimes \U_q(\mathfrak{sl}_{2r+2}).
$ This $\Delta$ will be called the [ coproduct]{} of ${\U^{\imath}}_q(\mathfrak{sl}_{2r+2})$, and $ \epsilon : {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \rightarrow \Qq$ will be called the [counit]{} of ${\U^{\imath}}_q(\mathfrak{sl}_{2r+2})$. The counit map $\epsilon$ makes $\Qq$ a (trivial) ${\U^{\imath}}_q(\mathfrak{sl}_{2r+2})$-module. Let $m : \U_q(\mathfrak{sl}_{2r+2}) \otimes \U_q(\mathfrak{sl}_{2r+2}) \rightarrow \U_q(\mathfrak{sl}_{2r+2})$ denote 