  In the present paper we continue the project of systematic construction of
invariant differential operators on the example of the non-compact algebras
$su(n,n)$. Earlier were given the main multiplets of indecomposable elementary
representations for $n\leq 4$, and the reduced ones for $n=2,3$. Here we give
all reduced multiplets containing physically relevant representations including
the minimal ones for $n=4$. Due to the recently established parabolic relations
the results are valid also for the algebras $sl(8,\mathbb{R})$ and $su^*(8)$
with suitably chosen maximal parabolic subalgebras.
