Consider the two-qubit Werner state, defined as $$\rho_z = z |\Psi_-\rangle\!\langle \Psi_-| + \frac{1-z}{4}I, \quad |\Psi_-\rangle\equiv\frac{1}{\sqrt2}(|00\rangle-|11\rangle),$$ for $z\ge0$. Using the PPT criterion, one can see that this state is separable iff $0\le z\le 1/3$. Equivalently, the same state can be written as a mixture over symmetric and antisymmetric (normalised) projections: $$\rho_p \equiv p \frac{\Pi_{\rm sym}}{\binom{d+1}{2}} + (1-p) \frac{\Pi_{\rm asym}}{\binom{d}{2}},$$ with $d=2$ for the two-qubit case, and $2\Pi_{\rm sym}\equiv I+\operatorname{SWAP}$, $2\Pi_{\rm asym}=I-\operatorname{SWAP}$. The two definitions are connected setting $p=\frac34 (1-z)$, and thus the separability threshold in this parameterisation is at $p=\frac12$.
I couldn't, however, find a source discussing explicit separable decompositions (in the $z\le1/3$ regime, of course). Is there a "nice" way to find such decompositions?