  Adapting the spectral viewpoint suggested in Ledoux (2012) in the context of
symmetric Markov diffusion generators and recently exploited in the
non-diffusive setup of a Poisson random measure by D\"obler and Peccati (2017),
we investigate the fourth moment condition for discrete multiple integrals with
respect to general, i.e.\ non-symmetric and non-homogeneous, Rademacher
sequences and show that, in this situation, the fourth moment alone does not
govern the asymptotic normality. Indeed, here one also has to take into
consideration the maximal influence of the corresponding kernel functions. In
particular, we show that there is no exact fourth moment theorem for discrete
multiple integrals of order $m\geq2$ with respect to a symmetric Rademacher
sequence. This behavior, which is in contrast to the Gaussian (see Nualart and
Peccati (2005)) and Poisson (see D\"obler and Peccati (2017)) situation,
closely resembles the conditions for asymptotic normality of degenerate,
non-symmetric $U$-statistics from the classical paper by de Jong (1990).
