= Solution
On the <tensor product> of spins $j_1$ and $j_2$, the combined <angular momentum operator> is $\mathbf J=\mathbf S^{(1)}\otimes I+I\otimes\mathbf S^{(2)}$. The <Clebsch-Gordan decomposition> is
$$
\boxed{\mathcal H_{j_1}\otimes\mathcal H_{j_2}\cong\bigoplus_{J=|j_1-j_2|}^{j_1+j_2}\mathcal H_J,}
$$
where $J$ increases by one and each <irreducible spin representation> occurs once. The <dimensions> agree: $\sum_J(2J+1)=(2j_1+1)(2j_2+1)$.
The uncoupled <orthonormal basis> diagonalizes $S_3^{(1)},S_3^{(2)}$; the coupled <orthonormal basis> diagonalizes $\mathbf J^2,J_3$. The <Clebsch-Gordan coefficients> are their change-of-<basis> entries, with convention
$$
|J,M\rangle=\sum_{m_1,m_2}C_{j_1,j_2}(J,M;m_1,m_2)\,|j_1,m_1\rangle\otimes|j_2,m_2\rangle.
$$
They vanish unless $m_1+m_2=M$. Since both <bases> are orthonormal, this change-of-<basis> <matrix> obeys <unitarity>. Overall phases of the coupled <basis> vectors are fixed by a convention and do not change the <energy eigenvalues>.
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