Its not elementary - its in fact quite deep.I thought this was elementary. Splits that support decoherence describe worlds such as the world opf Schrodinger Cats in which decoherence operates and a preferred basis emerges. Splits which do not support decoherence describe worlds in which decoherence does not operate such as polarization experiments. Clearly it would be silly to insist that decoherence is fundamental, when, in fact it depends on the factorization. This is why I say that decoherence is not fundamental to MWI, it is a welcome refinement.
Conceptually its simple. Consider the particle detector in Schrodinger Cat. I have posted a very elementary argument showing if you just observe the detector then because its entangled with the emitted particle its in a mixed state and MW, or any other decoherence based explanation for that matter, handles it easy peasy. But now lets decompose it into half the particle detector and the other half + the particle; do you still get the same result? Its difficult to analyse it that way and you need rocks in your head to do it, but we are speaking of matters of principle. We do not have theorems that show you will get the same result ie the same mixed state.