 Solution

In the regime of weak-field Zeeman splitting, $J,m_J$ remain good labels and $\Delta E={\mu }_B{g}_J{m}_{J}B$. For the 393.4 nm line, $J_u=3/2$ in $4p{}^2{P}_{3/2}$ and $J_l=1/2$ in $4s{}^2{S}_{1/2}$. The Landé g-factors are $g_u=4/3$ and $g_l\simeq 2$. Apply $\Delta m_J=0,\pm 1$ to the four upper and two lower sublevels. The allowed pairs $(m_u,m_l)$ areTheir frequency displacements in units of ${\mu }_{B}B/h$ are respectively $1,-1/3,-5/3,5/3,1/3,-1$, so none coincide. Looking perpendicular to the field permits both the $\pi$ radiation from $q=0$ transitions and the $\sigma$ radiation from $q=\pm 1$ transitions. ConsequentlyA linear polarizer parallel to the field isolates the two $\pi$ components; a transverse polarizer selects the four $\sigma$ components. Without polarization selection both sets are present. This count assumes a field small compared with fine structure coupling and neglects isotope-dependent hyperfine structure; the strong-field Paschen-Back effect or a resolved nonzero-spin isotope would require a different model. The spectral spacings are unnecessary for the count, but the distinct shifts verify that counting allowed transitions does not inadvertently count degenerate lines twice.