  We present strategies to derive Bell inequalities valid for systems composed
of many three-level parties. This scenario is formalized by a Bell experiment
with $N$ observers, each of which performs one out of two possible
three-outcome measurements on their share of the system. As the complexity of
the set of classical correlations prohibits its full characterization in this
multipartite scenario, we consider its projection to a lower dimensional
subspace spanned by permutationally invariant one- and two-body observables.
This simplification allows us to formulate two complementary methods for
detecting nonlocality in multipartite three-level systems, both having a
complexity independent of $N$. Our work can have interesting applications in
the detection of Bell correlations in paradigmatic spin-1 models, as well as in
experiments with solid-state systems or atomic ensembles.
