A complex Hadamard matrix of order $6$ is $H_2$-reducible if it is equivalent to a Hadamard matrix for which all the nine $2\times 2$ submatrices are Hadamard, see [72].
$H_2$-reducible matrices are easy to identify. A matrix $H$ is $H_2$-reducible, if and only if its dephased form has at least one element equal to $-1$.