= Solution
The four required <dot products> are
$$
\mathbf a_1\cdot\mathbf b_1=\frac1{\sqrt2},\quad
\mathbf a_1\cdot\mathbf b_3=-\frac1{\sqrt2},\quad
\mathbf a_3\cdot\mathbf b_1=\frac1{\sqrt2},\quad
\mathbf a_3\cdot\mathbf b_3=\frac1{\sqrt2}.
$$
Consequently the specified <CHSH inequality> combination is
$$
\boxed{S=-\frac1{\sqrt2}-\frac1{\sqrt2}-\frac1{\sqrt2}-\frac1{\sqrt2}=-2\sqrt2.}
$$
The magnitude reaches the quantum <Tsirelson bound> and exceeds the local bound two. The negative sign follows this singlet and the stated order of terms; other conventions can reverse it.
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