Matrix parallelogram laws and symmetric moduli

We revisit the classical parallelogram law and a quite recent one by Kittaneh–Losada–Mayyas. Several operator inequalities follow. For instance: given a matrix [Formula: see text] with Cartesian (Toeplitz) decomposition [Formula: see text], we have [Formula: see text] for some unitary matrices [Formula: see text]. A stronger inequality actually holds for the quadratic symmetric modulus and several related open questions are proposed. Hence the paper opens the way to a systematic study of inequalities for operator symmetric moduli.