= Solution
Use the electronic weak-field <Zeeman splitting> model, neglecting <hyperfine structure>. The <Landé g-factor> is $g_g=2$ for $3S_{1/2}$ and $g_e=2/3$ for $3P_{1/2}$. The shift of an absorption <frequency> is therefore
$$
\delta\nu=\frac{\mu_BB_z}{h}(g_em_e-g_gm_g),\qquad m_g,m_e=\pm\frac12.
$$
Here the <magnetic field> is weak relative to <spin-orbit coupling>, so $J$ remains a useful <quantum number>, but the <spectral line> widths are small enough to resolve its magnetic substates.
For <linear polarization> parallel to $z$, the optical <electric field> has only the field-axis $\pi$ component. The <electric-dipole selection rule> is $\Delta m=0$. Its two allowed transitions are $m_g=m_e=\pm1/2$, giving
$$
\boxed{\mathbf E\parallel\mathbf B:\qquad
\nu=\nu_0\pm\frac{2\mu_BB_z}{3h}.}
$$
For propagation perpendicular to $z$ with <linear polarization> also perpendicular to $z$, the optical <electric field> is an equal-amplitude combination of the two field-axis circular components. It drives $\sigma^\pm$ transitions with $\Delta m=\pm1$. These connect $-1/2\to+1/2$ and $+1/2\to-1/2$, giving
$$
\boxed{\mathbf E\perp\mathbf B:\qquad
\nu=\nu_0\pm\frac{4\mu_BB_z}{3h}.}
$$
Thus each orientation selects two lines, with twice the separation in the perpendicular-polarization case. Neither pair has an unshifted central component. This is <polarization-selective Zeeman splitting of an alkali D1 line>, an <anomalous Zeeman effect> rather than a normal orbital triplet.
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2019-q3-sodium-d1-lines.png]
{title=Polarization selects two different sodium D1 Zeeman doublets}
{description=Original electronic-level prediction in units $\mu_BB_z/h$. Equal heights within each pair illustrate an initially unpolarized lower level and a weak probe; optical pumping can alter populations and hence strengths. Hyperfine structure is omitted, as in the stated electronic $J$-level model.}
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