In this paper we explore in detail the symmetry of the effective non-Abelian gauge field model that characterize the system of twisted bilayer graphene (TBG). It is found that the total Lagrangian including the term of pseudo Rashba effect and the magnetic-like term is continuous locally U(1) invariant up to a total derivative term. The magnetic-like term is found that it is actually of the spin-orbit-interaction-like form which happens in the vector space of Lie algebra of the gauge transformation. The effective non-Abelian gauge fields are generalized as the time-dependent fields and the motion equation, which is analogous to the Yang-Mills equation except three extra current interaction terms, is achieved.