A is symmetric iff A = A^†. A diagonal matrix is symmetric. E and D are diagonal and symmetric: E = E^† and D = D^†. C = C^† by hypothesis. The product of symmetric matrices is not generally symmetric! (XY)^† = Y^†X^†.
(ECE)^† = ((EC)E)^† = E^†(EC)^† = E^†(C^†E^†) = E^†C^†E^† = ECE.
(ECE)^† = ECE.
Q.E.D.