  It is known previously that the operator product expansion (OPE) between the
first ${\cal N}=3 $ multiplet and itself contains the second ${\cal N}=3$
multiplet in the supersymmetric coset model. In this paper, by using their
realizations in terms of various fermions, we compute the four kinds of OPEs
between the first and the second ${\cal N}=3$ multiplets for fixed $N$ and $M$
where the group of the coset contains $SU(N+M)$. By supersymmetrizing the above
OPEs in ${\cal N}=3$ superspace and using the various Jacobi identities between
the currents, we determine the ${\cal N}=3$ supersymmetric OPE between the
first and the second ${\cal N}=3$ multiplets completely. The right hand side of
this OPE contains the various ${\cal N}=3$ multiplets: the $SO(3)$ singlet
${\cal N}=3$ multiplets of superspin-$\frac{3}{2},2,3,4$ and the $SO(3)$
triplet ${\cal N}=3$ multiplets of superspin-$\frac{5}{2},3,\frac{7}{2}$. The
${\cal N}=2$ superspace description and the decoupling of the
spin-$\frac{1}{2}$ current of the ${\cal N}=3$ superconformal algebra are also
described.
