Absolute term values for the quartet states of neutral lithium

New calculations on the core-excited $1s2s3s{}_{}{}^4{}_{}{}^{}S$ and $1s2p3d{}_{}{}^4{}_{}{}^{}D_{}^o{}_{}{}^{}$ states of neutral lithium combined with very accurate experimental and theoretical data confirm an earlier estimate of the position of the lowest quartet state, $1s2s2p{}_{}{}^4{}_{}{}^{}P_{\frac{3}{2}}^o{}_{}{}^{}$ at 57.4128(2) eV above the Li ground state. The uncertainty of the present result, ±0.2 meV, is five times smaller than our previous one and it arises, in part, from the uncertainty in the conversion factor from ${\mathrm{cm}}^{-1\mathrm{}}$ to eV. Good agreement with recent experimental work of Rassi et al. (57.41±0.015 eV) is obtained. The positions of all identified quartets have been recalculated from the latest experimental data and new theoretical results.