  Using the variational method, a detailed study of the lowest m = 0,-1
electronic states of the exotic molecular ion H_4^{3+} in a strong magnetic
field, in the linear symmetric configuration parallel to the direction of the
magnetic field is carried out. A extended study of the 1\sigma_g ground state
(J.C. L\'opez and A. Turbiner, Phys. Rev. A 62, 022510, 2000) was performed
obtaining that the potential energy curve displays a sufficiently deep minimum
for finite internuclear distances, indicating the possible existence of the
molecular ion H_4^{3+}, for magnetic fields of strength B \gtrsim 3x10^{13}G.
It is demonstrated that the excited state 1\pi_u can exist for a magnetic field
B = 4.414 x 10^{13}G corresponding to the limit of applicability of the
non-relativistic theory.
