  Let $\mathcal G$ be the group of all Interval Exchange Transformations.
Results of Arnoux-Fathi ([Arn81b]), Sah ([Sah81]) and Vorobets ([Vor17]) state
that $\mathcal G_0$ the subgroup of $\mathcal G$ generated by its commutators
is simple. In [Arn81b], Arnoux proved that the group $\overline{\mathcal G}$ of
all Interval Exchange Transformations with flips is simple.
  We establish that every element of $\overline{\mathcal G}$ has a commutator
length not exceeding $6$. Moreover, we give conditions on $\mathcal G$ that
guarantee that the commutator lengths of the elements of $\mathcal G_0$ are
uniformly bounded, and in this case for any $g\in \mathcal G_0$ this length is
at most $5$.
  As analogous arguments work for the involution length in $\overline{\mathcal
G}$, we add an appendix whose purpose is to prove that every element of
$\overline{\mathcal G}$ has an involution length not exceeding $12$.
