  We construct new pairs of orthogonal maximal abelian $*$-subalgebras of
$M_6(\mathbb C)$, by classifying all self-adjoint complex Hadamard matrices of
order 6. In particular, we exhibit a non-affine one-parameter family of
non-equivalent Hadamard matrices of order 6. In the last part of the paper we
present other previously unknown examples of complex Hadamard matrices of
higher orders.
