If $M$ is a Riemannian 4-symmetric manifold, then it is known that $M$ has three complex differentiable distributions $D_{-1}$, $D_1$ and $\overline D_1$ on it. We shall prove that there are three differentiable complementry projection operators $P$, $P_1$ and $\overline P_1$ on $M$ that project on $D_{-1}$, $D_1$ and $\overline D_1$ respectively. Some useful relations containing Nijenhuis tensor are found. Necessary and sufficient conditions for $D_{-1}$, $D_1$, and $\overline D_1$ to be integrable are studied.