= Solution
Represent spin by $J_i=(\hbar/2)\sigma_i$ in the basis $|\uparrow\rangle,|\downarrow\rangle$. The <Pauli matrices> are Hermitian and satisfy $[\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k$, giving the general <angular momentum commutation relations> $[J_i,J_j]=i\hbar\epsilon_{ijk}J_k$. The particular representation has $J^2=3\hbar^2I/4$ and $J_3$ eigenvalues $\pm\hbar/2$, the features specific to <spin one-half>.
In the stated order of two-particle basis vectors, the operator is
$$
\boxed{\sigma^{(A)}\cdot\sigma^{(B)}=\begin{pmatrix}1&0&0&0\\0&-1&2&0\\0&2&-1&0\\0&0&0&1\end{pmatrix}.}
$$
Its eigenvalue-one vectors are $|\uparrow\uparrow\rangle$, $|\downarrow\downarrow\rangle$ and $(|\downarrow\uparrow\rangle+|\uparrow\downarrow\rangle)/\sqrt2$, all symmetric under interchange. The eigenvalue-minus-three vector is $(|\downarrow\uparrow\rangle-|\uparrow\downarrow\rangle)/\sqrt2$, antisymmetric under interchange. These are the <triplet state> and <singlet state>, respectively.
Identical spin-half particles are <fermions>. With no other degrees of freedom to carry antisymmetry, \b[only the singlet is allowed]. Indeed, for total angular momentum $J=(\hbar/2)(\sigma^{(A)}+\sigma^{(B)})$,
$$
J^2=\frac{\hbar^2}4\left(6I+2\sigma^{(A)}\cdot\sigma^{(B)}\right).
$$
This gives $2\hbar^2$ on the triplet and zero on the singlet, corresponding to total spin one and zero, as required by <addition of angular momentum>.
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