Constructions of $SU(2)$ and Weyl equivariant maps for all classical groups

If $G$ is a compact Lie group, $T$ a maximal torus in $G$ (with Lie algebras $\mathfrak{g}$ and $\mathfrak{t}$ respectively) and $W$ the corresponding Weyl group, then the Berry-Robbins problem for $G$, as formulated by Sir Michael Atiyah and Roger Bielawski, asks whether there exists a continuous $SU(2) \times W$ equivariant map from the space of regular Cartan triples (an open subset of $\mathfrak{t} \otimes \mathbb{R}^3$) to $G/T$, where $SU(2)$ acts via a regular Lie group homomorphism $SU(2) \to G$. This was settled positively by Atiyah and Bielawski, and their maps are even smooth, but they are not explicit. For $G=U(n)$, there exists another construction due to Sir Michael Atiyah and developed further with Paul Sutcliffe, which is explicit, but relies on a linear independence conjecture. The author had previously found a similar type of construction for $G=Sp(m)$, also relying on a linear independence conjecture. In this paper, similar constructions are done for $SO(2m+1)$ and $SO(2m)$, thus exhausting the list of classical groups.