  We discuss here how, when higher-order effects in the parameter
$\frac{\ell}{\hbar} |\vec{p}|$, related to the lattice spacing $\ell$, are
considered, pristine graphene, and other Dirac materials, can be used as
tabletop systems where generalized commutation relations are naturally
realized. Such generalized algebras of quantization, which lead to generalized
versions of the Heisenberg uncertainty principle, are under intense scrutiny
these days, as they could manifest a fundamental length scale of spacetime.
Despite the efforts and the many intriguing results, there are no experimental
signatures of any generalized uncertainty principle (GUP). Therefore, our
results here, which tell how to use tabletop physical systems to test certain
GUPs in analog experiments, should be of interest to practitioners of quantum
gravity. We identify three different energy regimes that we call ``layers'',
where the physics is still of a Dirac type but within precisely described
limits. The higher the energy, the more sensitive the Dirac system becomes to
the effects of the lattice. Here such lattice plays the role of a discrete
space where the Dirac quasi-particles live. With the goals just illustrated, we
had to identify the mapping between the high-energy coordinates, $X^i$, and the
low-energy ones, $x^i$, i.e., those measured in the lab. We then obtained three
generalized Heisenberg algebras. For two of them we have the noticeable result
that $X^i = x^i$, and for the third one we obtained an improvement with respect
to an earlier work: the generalized coordinates expressed in terms of the
standard phase space variables, $X^i(x,p)$, and higher order terms.
