symmetric pairs shall be given in the first two sections, separately (see also [@ES13]). The theory of $\imath$-canonical bases are nevertheless uniform in both cases. Therefore after stating their definitions we shall formulate their general theory together. Section $2.1$ - Section $2.3$ are analogous to [@BW13 Part 1] hence we shall omit the proofs almost entirely. Some of the quantum symmetric paris considered here are of different parameters than the ones considered in [@BW13] (See also [@BW16] for more general construction). We refer to [@Ko14] for general theory of quantum symmetric pairs.

The quantum symmetric pair $(\U_q(\mathfrak{sl}_{2r+1}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}))$ {#sec:Ui}
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We define $$\I^{\imath} _{2r} = (\hf +\N) \cap \I_{2r} = \Big\{\hf, \frac32, \ldots, r-\hf \Big\}.$$ The Dynkin diagram of type $A_{2r}$ together with the involution $\inv$ are depicted as follows:

(-1.5,0) node [$A_{2r}:$]{}; (0.5,0) node\[below\] [$\alpha_{-r+\hf}$]{} – (2.5,0) node\[below\] [$\alpha_{-\hf}$]{} ; (2.5,0) – (3.5,0) node\[below\] [$\alpha_{\hf}$]{}; (3.5,0) – (5.5,0) node\[below\] [$\alpha_{r-\hf}$]{} ; (0.5,0) node (-r) [$\bullet$]{}; (2.5,0) node (-1) [$\bullet$]{}; (3.5,0) node (1) [$\bullet$]{}; (5.5,0) node (r) [$\bullet$]{}; (-r.north east) .. controls (3,1) .. node\[above\] [$\theta$]{} (r.north west) ; (-1.north) .. controls (3,0.5) .. (1.north) ;

The algebra ${\U^{\imath}}_q(\mathfrak{sl}_{2r+1})$ is defined to be the associative algebra over $\Q(q)$ generated by 