In the time-dependent electro-nuclear formulation of the exact factorization, the development of approximations to compute the "effective potentials" of the theory, usually referred to as time-dependent potential energy surface and time-dependent vector potential, for molecules has been achieved after assessing their behavior on prototypical case studies where the exact numerical solution of the problem was accessible . Following the successful strategy of the electro-nuclear formulation, in this paper, we first reconstruct the time-independent exact electron factorization formalism in a general case manifesting a non-vanishing electronic vector potential and then we analyze such vector potential on a prototypical case study. Adopting an exactly-solvable two-electron non-interacting problem, we aim to characterize the most relevant features of the electronic vector potential, in particular demonstrating its relation to the property of the electronic wavefunction of being complex and showing that it cannot be gauged away (thus it must be accounted for in an approximate treatment of the problem).