  We derive the relativistic factor for splitting of the $g$-factors of a
fermion and its anti-fermion partner, which is important for placing
constraints on dimension-5, $CPT$-odd and Lorentz-invariance-violating
interactions from experiments performed in a cyclotron. From existing data, we
extract limits (1$\sigma$) on the coupling strengths of the temporal component,
$f^0$, of a background field (including the field amplitude), which is
responsible for such $g$-factor splitting, with an electron, proton, and muon:
$|f^0_e|< 2.3 \times 10^{-12} ~\mu_{\textrm{B}}$, $|f^0_p|< 4 \times 10^{-9}
~\mu_{\textrm{B}}$, and $|f^0_\mu|< 8 \times 10^{-11} ~\mu_{\textrm{B}}$,
respectively, in the laboratory frame. From existing data, we also extract
limits on the coupling strengths of the spatial components, $d^{\perp}$, of
related dimension-5 interactions of a background field with an electron,
proton, neutron, and muon: $| {d}_e^{\perp} | \lesssim 10^{-9}
~\mu_{\textrm{B}}$, $| {d}_p^{\perp} | \lesssim 10^{-9} ~\mu_{\textrm{B}}$, $|
{d}_n^{\perp} | \lesssim 10^{-10} ~\mu_{\textrm{B}}$, and $| {d}_\mu^{\perp} |
\lesssim 10^{-9} ~\mu_{\textrm{B}}$, respectively, in the laboratory frame.

High Energy Physics - Phenomenology;Nuclear Theory;Atomic Physics;Space Physics;Quantum Physics