Paradoxically, using Ariel cured DD's eczema, and I hit on this quite by accident.
I was using all sorts of non bio washing powders and liquids, and eco eggs, and the eczema got worse. The washing machine started to stink as well.
On a mumsnet thread someone posted a link to this website. Clearly my washing machine had developed quite a lot of mould and bacteria build up, and it was this that was causing the eczema.
Using a bio powder (not liquid) with a few hot temperature washes killed off the mould and bacteria. The result was a smell free washer, and another happy outcome was that DD's skin cleared up.
It isn't the enzymes in bio powder that cause the eczema, but the fragrance that is added to it, and washing at low temperatures.
I now use Ariel bio for whites and lights, and Fairy non bio for colours. I also use Fairy conditioner. I also wash bedding and towels at 60 degrees.