The sorting index and set-valued joint equidistributions of ${\mathcal{B}}_n$ and ${\mathcal{D}}_n$

The sorting indices ${\text{sor}}_B$ and ${\text{sor}}_D$ on the Coxeter groups of type $B$ and $D$ respectively are defined by Petersen and it is proved that $({\text{inv}}_B,\text{rlmin})$ and $({\text{sor}}_B,{\mathrm{\ell }}_B^{\mathrm{\prime }})$ have the same joint distribution for type $B$ while ${\text{inv}}_D$ and ${\text{sor}}_D$ have the same distribution for type $D$. These results, including a set-valued extension of type $B$ involving two equildistributed pairs of three statistics, are proved combinatorially by Chen et al. via two mappings $\phi :={\text{(B-code)}}^{-1}\circ \text{(A-code)}$ and $\psi :={\text{(D-code)}}^{-1}\circ \text{(C-code)}$. In this paper we further extend these results. In type $B$ we prove a set-valued joint equildistribution between a pair of seven statistics, and find a five-variable generating function. In type $D$ we define new set-valued statistics, among them ${\text{Cyc}}_D^{+}$ and ${\text{Cyc}}_D^{-}$, and firstly find a set-valued joint equidistribution between a pair of five statistics and find a four-variable generating function.