  Let $q$ be a power of a prime $p$, let $G$ be a finite Chevalley group over
$\mathbb{F}_q$ and let $U$ be a Sylow $p$-subgroup of $G$; we assume that $p$
is not a very bad prime for $G$. We explain a procedure of reduction of
irreducible complex characters of $U$, which leads to an algorithm whose goal
is to obtain a parametrization of the irreducible characters of $U$ along with
a means to construct these characters as induced characters. A focus in this
paper is determining the parametrization when $G$ is of type $\mathrm{F}_4$,
where we observe that the parametrization is "uniform" over good primes $p >
3$, but differs for the bad prime $p = 3$. We also explain how it has been
applied for all groups of rank $4$ or less.
