  We propose a very simple toy model of a $\mathbb{Z}_2^2$-supersymmetric
quantum system and show, via Klein's construction, how to understand the system
as being an $N=2$ supersymmetric system with an extra $\mathbb{Z}_2^2$-grading.
That is, the commutation/anticommutation rules are defined via the standard
boson/fermion rules, but the system still has an underlying
$\mathbb{Z}_2^2$-grading that needs to be taken into account.
