https://imgur.com/a/0r18d5R
More useful set equivalence goodness:
The diagonally adjacent cells are contained in four-square blocks arranged in a sort of x-wing pattern. In the above example, the purple squares are either housed in the blue group or the red group and either the green group or the orange group. All of the cells in the blue group in rows 3 and 4 are contained in the killer cage and therefore cannot be diagonally adjacent. That means that purple is in the red group. That also means by block 3 that the other 4 purples are housed in the orange group.
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