For SU(4), local equivalence classes under SU(2)⊗SU(2) multiplication are not geometrically represented by the Weyl chamber; that chamber appears only under projective-local equivalence that ignores global phases.
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On the KAK Decomposition and Equivalence Classes
For SU(4), local equivalence classes under SU(2)⊗SU(2) multiplication are not geometrically represented by the Weyl chamber; that chamber appears only under projective-local equivalence that ignores global phases.