 Solution

Using $T=2\mathrm{SWAP}-I$, the symmetric triplet Bell state have eigenvalue $1$, while the antisymmetric singlet has eigenvalue $-3$. With the ordering shown, ${\mathcal{B}}_1,{\mathcal{B}}_2,{\mathcal{B}}_3$ are triplet combinations and ${\mathcal{B}}_4$ is the singlet.