In the present paper we continue the project of systematic
construction of invariant differential operators on the example of
the non-compact algebra $so^*(8) \cong so(6,2)$. We use the maximal Heisenberg parabolic subalgebra ${\cal P} = {\cal M} \oplus {\cal A} \oplus {\cal N}$ with ${\cal M} = so^*(4)\oplus so(3)$. We give the main multiplets of indecomposable elementary
representations (ERs). This includes the explicit parametrization of the invariant differential operators between the ERS. \\
Due to the recently established parabolic relations the multiplet classification results are valid also for the two algebras $so(p,q)$ (for $(p,q)=(5,3), (4,4)$) with maximal Heisenberg parabolic
subalgebra: ${\cal P}' = {\cal M}' \oplus {\cal A}' \oplus {\cal N}'$, ${\cal M}' = so(p-2,q-2)\oplus sl(2,R)$, ${\cal M}'^C \cong {\cal M}^C.