  Flat bands (FBs) in materials can enhance the correlation effects, resulting
in exotic phenomena. Line graph (LG) lattices are well known for hosting FBs
with isotropic hoppings in $s$-orbital models. Despite their prevalent
application in the Kagome metals, there has been a lack of a general approach
for incorporating higher-angular-momentum orbitals with spin-orbit couplings
(SOCs) into LGs to achieve FBs. Here, we introduce a non-Abelian LG theory to
construct FBs in realistic systems, which incorporates internal degrees of
freedom and goes beyond $s$-orbital models. We modify the lattice edges and
sites in the LG to be associated with arbitrary Hermitian matrices, referred to
as the multiple LG. A fundamental aspect involves mapping the multiple LG
Hamiltonian to a tight-binding (TB) model that respects the lattice symmetry
through appropriate local non-Abelian transformations. We establish the general
conditions to determine the local transformations. Based on this mechanism, we
demonstrate the realization of $d$-orbital FBs in the Kagome lattice, which
could serve as a minimal model for understanding the FBs in transition metal
Kagome materials. Our approach bridges the gap between the known FBs in pure
lattice models and their realization in multi-orbital systems.
