Spin–lattice relaxation rate divided by temperature, 1/T₁T, under strong magnetic field is calculated in the tilted Weyl Hamiltonian with the long range Coulomb interaction, which describes the low energy electronic states in organic conductor α-(BEDT-TTF)₂I₃. Effects of the spin and valley orders in the N = 0 Landau levels are treated by the mean field theory. Effects of the fluctuations corresponding to those orders are taken into account using the ladder vertex of the long range Coulomb interaction. It is shown that 1/T₁T has a shoulder structure due to the N = 0 Landau levels with the spin splitting, where the velocity renormalization and the enhancement of the effective g-factor is required to adjust the present numerical results to the NMR measurement. It is also shown that an upturn of 1/T₁T at low temperatures is induced by the spin transverse fluctuation enhanced by the ladder vertex, although the upturn cannot be explained by the valley splitting.

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