  It is shown that a normalized complex Hadamard matrix of order $6$ having
three distinct columns, each containing at least one $-1$ entry necessarily
belongs to the transposed Fourier family, or to the family of $2$-circulant
complex Hadamard matrices. The proofs rely on solving polynomial system of
equations by Gr\"obner basis techniques, and make use of a structure theorem
concerning regular Hadamard matrices. As a consequence, members of these two
families can be easily recognized in practice. In particular, one can identify
complex Hadamard matrices appearing in known triplets of pairwise mutually
unbiased bases in dimension $6$.
