Eczema from detergents – per solveeczema.org – is caused by how detergents affect skin membrane permeability and function and is not a true IgE-mediated allergy to detergents. In fact, very few people (if any) have a true allergy to detergents, most detergent allergies are actually allergies to product enzymes and additives. Sodium lauryl sulfate, known to be highly problematic for people with eczema, is a detergent analog of a surfactant our bodies make and use to regulate membrane permeability, among other things. I’m not sure we even could be allergic to it in the true sense; again, the eczema comes about from how it affects the skin membrane.
Thus you could not test for detergent reactivity with traditional allergy blood tests. Initially I sometimes used the word “allergy” in regards to this reaction, with some justification, but it’s not a true allergic reaction to the detergents themselves.