  We present an algorithm for the classification of triples of lattice
polytopes with a given mixed volume $m$ in dimension 3. It is known that the
classification can be reduced to the enumeration of so-called irreducible
triples, the number of which is finite for fixed $m$. Following this algorithm,
we enumerate all irreducible triples of normalized mixed volume up to 4 that
are inclusion-maximal. This produces a classification of generic trivariate
sparse polynomial systems with up to 4 solutions in the complex torus, up to
monomial changes of variables. By a recent result of Esterov, this leads to a
description of all generic trivariate sparse polynomial systems that are
solvable by radicals.
