= Solution
The coupled <basis> consists of the triplet
$$
|1,1\rangle=|++\rangle,
\quad |1,0\rangle=\frac{|+-\rangle+|-+\rangle}{\sqrt2},
\quad |1,-1\rangle=|--\rangle,
$$
and the singlet
$$
|0,0\rangle=\frac{|+-\rangle-|-+\rangle}{\sqrt2}.
$$
These equations list all nonzero Clebsch--Gordan coefficients, each coefficient being $1$, $1/\sqrt2$, or $-1/\sqrt2$ as displayed, and
$$
I=\sum_{m=-1}^{1}|1,m\rangle\langle1,m|+|0,0\rangle\langle0,0|.
$$
Because each irreducible total-spin representation occurs once, an operator commuting with total <angular momentum> is
$$
O=\alpha P_{j=1}+\beta P_{j=0}.
$$
In the product <basis>, the nonzero <matrix> elements are
$$
\langle++|O|++\rangle=\langle--|O|--\rangle=\alpha,
$$
$$
\langle+-|O|+-\rangle=\langle-+|O|-+\rangle=\frac{\alpha+\beta}{2},
$$
and
$$
\langle+-|O|-+\rangle=\langle-+|O|+-\rangle=\frac{\alpha-\beta}{2}.
$$
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