\bff_{\alpha_1}\bt^2 &= (q + q^{-1}) \bt\bff_{\alpha_1}\bt + \bff_{\alpha_1}.\displaybreak[0]\end{aligned}$$ We introduce the divided powers $\be^{(a)}_{\alpha_i} = \be^a_{i} / [a]!$, $\bff^{(a)}_{\alpha_i} = \bff^a_{i} / [a]!$ for $a \ge 0$, $i \in \I^{\imath}_{2r+1}$.

\[rem:Uisame\] The generating relations of the algebra ${\U^{\imath}}_q(\mathfrak{sl}_{2r+2})$ are the same as the generating relations of the algebra considered in [@BW13 §2.1].

\[lem:3inv\] The $\Q$-algebra ${\U^{\imath}}_q(\mathfrak{sl}_{2r+2})$ has an anti-linear ($q \mapsto q^{-1}$) bar involution such that $\ov{\bk}_{\alpha_i} = \bk^{-1}_{\alpha_i}$, $\ov{\be}_{\alpha_i} = \be_{\alpha_i}$, $\ov{\bff}_{\alpha_i} = \bff_{\alpha_i}$, and $\ov{\bt} = \bt$ for all $i \in \I^{\imath}_{2r+1}$.

(Sometimes we denote the bar involution on ${\U^{\imath}}_q(\mathfrak{sl}_{2r+2})$ by $\Bbar$.)

\[prop:embedding\] There is an injective $\Qq$-algebra homomorphism $\imath :  {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \rightarrow \U_q(\mathfrak{sl}_{2r+2})$ which sends (for $i \in \I^{\imath}_{2r+1}$) $$\begin{aligned}
\bk_{\alpha_i} \mapsto K_{\alpha_i}K^{-1}_{\alpha_{-i}}, \qquad & \bt \mapsto E_{\alpha_0} +qF_{\alpha_0}K^{-1}_{\alpha_0} \\
 \be_{\alpha_i} \mapsto  E_{\alpha_i} + K^{-1}_{\alpha_i}F_{\alpha_{-i}},\qquad 
& \bff_{\alpha_i} \mapsto F_{\alpha_i} K^{-1}_{\alpha_{-i}}+ E_{\alpha_{-i}}.\end{aligned}$$

The embedding $\imath$ in Proposition \[prop:embedding\] is different from the embedding in [@BW13 Proposition 2.2], although the two subalgebras are (abstractly) isomorphic (see Remark \[rem:Uisame\]). This phenomenon for quantum symmetric pairs was first observed in [@Le Section 5].

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 all $ 0 \neq i \in \I$. Using the quantum binomial formula [@Lu94 1.3.5], we have, for all $i 