= Solution
Use orthonormal basis <vectors> $\mathbf e_i(t)$ fixed in the <rotating reference frame>, and expand the <angular momentum> as $\mathbf J=\sum_iJ_i\mathbf e_i$. Over time $dt$, the basis undergoes infinitesimal <rotation> $\boldsymbol\Omega\,dt$, so
$$
\left(\frac{d\mathbf e_i}{dt}\right)_{S_0}=\boldsymbol\Omega\times\mathbf e_i.
$$
The <product rule> gives the <rotating-frame derivative formula>:
$$
\left(\frac{d\mathbf J}{dt}\right)_{S_0}
=\sum_i\dot J_i\mathbf e_i+\sum_iJ_i\boldsymbol\Omega\times\mathbf e_i
=\left(\frac{d\mathbf J}{dt}\right)_S+\boldsymbol\Omega\times\mathbf J.
$$
Since the applied <couple> is the rate of change of <angular momentum> in the <inertial frame>,
$$
\boxed{\mathbf G=\left(\frac{d\mathbf J}{dt}\right)_S+\boldsymbol\Omega\times\mathbf J.}
$$
The second term accounts for the changing basis, even when the components of <angular momentum> are constant in the <rotating reference frame>.
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