$\be_{\alpha_i}$, $\bff_{\alpha_i}$, $\bk_{\alpha_i}$, $\bk^{-1}_{\alpha_i}$, $i \in \I^{\imath} _{2r} $, subject to the following relations for $i, j \in \I^{\imath} _{2r} $: $$\begin{aligned}
 \ibk{i} \ibk{i}^{-1} &= \ibk{i}^{-1} \ibk{i} =1, \displaybreak[0] \notag\\
  \ibk{i} \ibk{j} &=  \ibk{j}  \ibk{i}, \displaybreak[0] \notag\\
  \ibk{i} \ibe{j} \ibk{i}^{-1} &= q^{(\alpha_i-\alpha_{-i},\alpha_j)} \ibe{j}, \displaybreak[0] \notag\\
  \ibk{i} \ibff{j} \ibk{i}^{-1} &= q^{-(\alpha_i-\alpha_{-i},\alpha_j)}\ibff{j}, \displaybreak[0] \notag\\
  \be_{\alpha_i} \ibff{j} -\ibff{i} \ibe{j} &= \delta_{i,j} \frac{\bk_{\alpha_i}
 -\bk^{-1}_{\alpha_i}}{q-q^{-1}},         \qquad\; \qquad \text{if } i, j \neq \hf, \displaybreak[0] \notag\\
  \ibe{i}^2 \ibe{j} +\ibe{j} \ibe{i}^2 &= (q+q^{-1}) \ibe{i} \ibe{j} \ibe{i},  \qquad \text{if }  |i-j|=1, \displaybreak[0] \notag \\
  \ibff{i}^2 \ibff{j} +\ibff{j} \ibff{i}^2 &= (q+q^{-1}) \ibff{i} \ibff{j} \ibff{i}, \qquad \text{if } |i-j|=1,\displaybreak[0]\notag \\
  \ibe{i} \ibe{j} &= \ibe{j} \ibe{i},     \quad\qquad\qquad\qquad\; \text{if } |i-j|>1, \displaybreak[0]\notag \\
  \ibff{i} \ibff{j}  &=\ibff{j}  \ibff{i},   \quad\qquad\qquad\qquad\; \text{if } |i-j|>1,\displaybreak[0]\notag \\
    \ibff{\hf}^2\ibe{\hf} + \ibe{\hf}\ibff{\hf}^2
    &= (q+q^{-1}) \Big(\ibff{\hf}\ibe{\hf}\ibff{\hf}-q^2\ibff{\hf}\ibk{\hf}^{-1}-q^{-2}\ibff{\hf}\ibk{\hf} \big),\displaybreak[0] \label{eq:Serre:1}\\
  \ibe{\hf}^2\ibff{\hf} + \ibff{\hf}\ibe{\hf}^2
   &= (q+q^{-1}) \Big(\ibe{\hf}\ibff{\hf}\ibe{\hf}-q^{-2}\ibk{\hf}\ibe{\hf} -q^2\ibk{\hf}^{-1}\ibe{\hf} \Big).\displaybreak[0] \label{eq:Serre:2}\end{aligned}$$ We introduce the divided powers $\be^{(a)}_{\alpha_i} = \be^a_{\alpha_i} / [a]!$, $\bff^{(a)}_{\alpha_i} = \bff^a_{\alpha_i} / [a]!$.

Note that the last two “Serre" type relations and are different from [@BW13 §6.1].

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

(Sometimes we denote the 