Separable elements and splittings in Weyl groups of Type $B$

Separable elements in Weyl groups are generalizations of the well-known class of separable permutations in symmetric groups. Gaetz and Gao showed that for any pair $(X,Y)$ of subsets of the symmetric group ${\mathfrak{S}}_n$, the multiplication map $X\times Y\to {\mathfrak{S}}_n$ is a splitting (i.e., a length-additive bijection) of ${\mathfrak{S}}_n$ if and only if $X$ is the generalized quotient of $Y$ and $Y$ is a principal lower order ideal in the right weak order generated by a separable element. They conjectured this result can be extended to all finite Weyl groups. In this paper, we classify all separable and minimal non-separable signed permutations in terms of forbidden patterns and confirm the conjecture of Gaetz and Gao for Weyl groups of type $B$.