How can we be sure of anything?
Fred likes to remember things his own way, not necessarily how they actualy happened.
Practically the whole movie plays within Fred's own mind. The ONLY thing we realy can be sure of is that Fred is convicted and tried for the murder of his wife, because that segment plays out in reality.
We realy don't know WHY Fred killed his wife. Sure, we see him being jealous and thinking she has an affair. But did she realy? And was Fred indeed unable to sexualy satisfy her? The point is: these may be just excuses for Fred to justify his murder. By this I mean that he tries to diminish his guilt. We simply don't know what realy happened, but we DO know that we cannot trust Fred's own memory, because he himself says so.
I don't believe Fred killed Mr. Eddy / Dick Laurent for several reasons. In fact we cannot be sure Mr. Eddy / Dick Laurent even existed! He can simply be an excuse for Fred to justify his deed. The guards in the jailhouse refer to him as a wife killer, not as a serial killer. The cops confront Fred with the murder of his wife, not another murder.
So is there anything tangible for the viewer to go on?