  We revisit 2D $\mathcal{N}=(2,2)$ super Yang-Mills lattice formulation
(Sugino model) to investigate its fermion action with two (Majorana) fermion
flavors and exact chiral-$U(1)_{R}$ symmetry. We show that the reconcilement of
chiral symmetry and absence of further species-doubling originates in the 4D
clifford algebra structure of the action, where 2D two flavors are spuriously
treated as a single 4D four-spinor with four 4D gamma matrices introduced into
kinetic and Wilson terms. This fermion construction based on the
higher-dimensional clifford algebra is extended to four dimensions in two
manners: (1) pseudo-8D sixteen-spinor treatment of 4D four flavors with eight
8D gamma matrices, (2) pseudo-6D eight-spinor treatment of 4D two flavors with
five out of six 6D gamma matrices. We obtain 4D four-species and two-species
lattice fermions with unbroken subgroup of chiral symmetry and other essential
properties. We discuss their relations to staggered and Wilson twisted-mass
fermions. We also discuss their potential feedback to 4D super Yang-Mills
lattice formulations.
