H₂-reducible Hadamard matrices of order 6

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Complex Hadamard matrices H of order 6 are characterized in a novel manner, according to the presence/absence of order 2 Hadamard submatrices. It is shown that if there exists one such submatrix, H is equivalent to a Hadamard matrix where all the nine submatrices are Hadamard. The ensuing subset of H₂-reducible complex Hadamard matrices is more general than might be thought, and, significantly, includes all the up till now described (one- and two-parameter) families of order 6. A known, isolated matrix, and most numerically generated matrices, fall outside the subset.