  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 $X\times Y\rightarrow \mathfrak{S}_n$ 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
$B$.
