= Solution
The <vector triple product identity> is
$$
\mathbf a\times(\mathbf b\times\mathbf c)
=\mathbf b(\mathbf a\cdot\mathbf c)-\mathbf c(\mathbf a\cdot\mathbf b).
$$
Consequently a choice of the requested <coefficients> is
$$
\boxed{\alpha=0,\qquad\beta=\mathbf a\cdot\mathbf c,\qquad\gamma=-\mathbf a\cdot\mathbf b}.
$$
These are the standard identity's <coefficients>, even if the particular three <vectors> are <linearly dependent> and a representation need not be unique.
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