Every regular dihedral equiangular tight frame of redundancy 2 is genuinely projective and corresponds exactly to a 2-negacirculant skew Hadamard matrix.
Davidson, C*-algebras by example , Fields Institute for Research in Mathematical Sciences Toronto: Fields Institute monographs, American Mathematical Society, 1996
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Abelian and Dihedral equiangular tight frames of redundancy $2$
Every regular dihedral equiangular tight frame of redundancy 2 is genuinely projective and corresponds exactly to a 2-negacirculant skew Hadamard matrix.