We analyze the fulfillment of the simultaneous diagonalization (SD via congruence) property for any two real matrices, and develop sufficient conditions expressed in different way to those appeared in the last few years. These conditions are established under a different perspective, and in any case, they supplement and clarify other similar results published elsewhere. Following our point of view reflected in a previous work, we offer some necessary and sufficient conditions, different in nature to those in Jiang and Li (SIAM J Optim 26:1649-1668, 2016), for SD: roughly speaking our approach is more geometric and needs to compute images and kernels of matrices; whereas that in Jiang and Li (SIAM J Optim 26:1649-1668, 2016) requires to compute determinant and canonical forms. The bidimensional situation is particularly analyzed, providing new more precise characterizations than those in higher dimension and joint those given earlier by the authors. In addition, we also establish the connection of our characterization of SD with that provided in Jiang and Li (SIAM J Optim 26:1649-1668, 2016).