  We study one-dimensional systems of two-orbital SU(4) fermionic cold atoms.
In particular, we focus on an SU(4) spin model [named SU(4) $e$-$g$ spin model]
that is realized in a low-energy state in the Mott insulator phase at the
filling $n_g=3, n_e=1$ ($n_g, n_e$: numbers of atoms in ground and excited
states, respectively). Our numerical study with the infinite-size density
matrix renormalization group shows that the ground state of SU(4) $e$-$g$ spin
model is a nontrivial symmetry protected topological (SPT) phase protected by
$Z_4 \times Z_4$ symmetry. Specifically, we find that the ground state belongs
to an SPT phase with the topological index $2\in\mathbb{Z}_4$ and show sixfold
degenerate edge states. This is topologically distinct from SPT phases with the
index $1\in\mathbb{Z}_4$ that are realized in the SU(4) bilinear model and the
SU(4) Affleck-Kennedy-Lieb-Tasaki (AKLT) model. We explore the phase diagram of
SU(4) spin models including $e$-$g$ spin model, bilinear-biquadratic model, and
AKLT model, and identify that antisymmetrization effect in neighboring spins
(that we quantify with Casimir operators) is the driving force of the phase
transitions. Furthermore, we demonstrate by using the matrix product state how
the $\mathbb{Z}_4$ SPT state with six edge states appears in the SU(4) $e$-$g$
spin model.
