We may be at cross purposes here, both speaking of different substances often called by the same term. I don't have a dishwasher, other than any free pair of hands to take their turn, therefore when speak of dishwashing liquid I mean the one used to do dishes by hand in the sink. To me this is the one that properly deserves the term since it's the first way everyone did their dishes. You SHOULD NOT mix that one with Chlorine based products or any other bleach. I just took a look on the Ajax bottle and it specifically says to not use with any bleaching product. Check your bottle of HAND dishwashing liquid and see if it says the same.
There is another "dishwashing liquid" that has very different chemical properties, often has bleach already in it and can have more bleach added to it, and that is normally the type used in dishwashing MACHINES.
It would have been helpful if they'd come up with a different term for the soap or alkaline based chemicals used in dishwashing machines, instead of also calling it dishwashing liquid or powders.
Here's a link and short blurp about the products.
DISHWASHING HAND LIQUIDS : Usually contains ammonia; ethanol; formaldehyde (a potential human carcinogen), and glycol ethers. Exposure to fumes may cause labored breathing or coughing; intense sneezing; or, in some cases, death. Direct contact with eyes may cause tearing and puffiness around eyes. Direct contact with skin may cause severe irritation. Mixing dishwashing liquids with BLEACH (for sanitizing purposes) can produce "toxic", chloramine fumes. Repeated exposure may cause asthma-like symptoms and/or effect the central nervous system over a long period of time. Label precautions read "CAUTION"
DISHWASHING LIQUIDS, AUTOMATIC : Usually contain bleaching agents; artificial dyes ( see "DYES" ); ethyl alcohol; and artificial fragrances ( see "FRAGRANCES" ). Undiluted, may act as a strong skin irritant. If swallowed, may burn your throat . In those products containing bleaching agents, chloramine gas may be created whose fumes may cause swelling and tearing of eyes; or even, faintness. Over a prolonged period of time, these fumes may cause headaches; lung irritation; nausea; and depression. In extreme cases, prolonged exposure may even cause coma and possibly, death! Label precautions read "WARNING" !
DISHWASHING POWDERS, AUTOMATIC :
THE MOST REPORTED POISON HARMING OR KILLING YOUNG CHILDREN.
Usually contains sodium silicate and/or bleach. Both ingredients are strong eye and skin irritants. Bleach content is CORROSIVE and may burn the mouth and throat. Repeated and/or prolonged exposure to vapors or dust may cause lung irritation, central nervous system disorders, and in severe cases, death! Label precautions read "WARNING" !