We propose a very simple toy model of a Z₂²-supersymmetric quantum system and show, via Klein's construction, how to understand the system as being an $N=2$ supersymmetric system with an extra Z₂²-grading. That is, the commutation/anticommutation rules are defined via the standard boson/fermion rules, but the system still has an underlying Z₂²-grading that needs to be taken into account.
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Andrew James Bruce (2024) studied this question.
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