= Solution
Within a fixed isolated electronic $J$ multiplet, the electronic field at the <nucleus> is a vector operator. <Rotational symmetry> and the <Wigner-Eckart theorem> imply that its projection into this multiplet is proportional to $\mathbf J$:
$$
P_J\mathbf B_eP_J=b_J\mathbf J.
$$
Its detailed orbital, spin-dipolar and contact contributions are included in the reduced coefficient $b_J$. Consequently the projected <magnetic-dipole hyperfine interaction> becomes
$$
\boxed{H_{\rm hfs}=A\mathbf I\cdot\mathbf J,\qquad A=-g_I\mu_Nb_J,}
$$
in the same dimensionless angular-momentum convention as <2/a>. The proportionality is a statement about <matrix elements> \b[within] the chosen electronic manifold, not an identity for the <electron> field operator on the entire atomic Hilbert space.
Define $\mathbf F=\mathbf I+\mathbf J$. In zero external field the <quantum states> obtained by <addition of angular momentum> are labelled $|\gamma J,I;F,m_F\rangle$, where $\gamma$ specifies the <electron configuration> and other fixed labels. The good quantum numbers include $I,J,F,m_F$, with $F=|I-J|,\ldots,I+J$ and $m_F=-F,\ldots,F$. In a description using <LS coupling> the fixed electronic labels can also include $L,S$. The <energies> depend on $F$ but not $m_F$:
$$
E_F=\frac A2[F(F+1)-I(I+1)-J(J+1)].
$$
This neglects coupling to other electronic manifolds, <electric-quadrupole hyperfine interactions> and the <Zeeman effect> from external fields. Those additions can shift the intervals or change which quantum numbers remain useful.
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