After recall classical results in order to place our work, we introduce the class of split regular BiHom-Lie algebras as the natural extension of the one of split Hom-Lie algebras and so of split Lie algebras. By making use of connection techniques, we focus our attention on the study of the structure of such algebras and, under certain conditions, the simplicity is characterized. Finally, it is shown that a split regular BiHom-Lie algebra is the direct sum of the family of its simple ideals, each one being a simple split regular BiHom-Lie algebra with a root system having all its non-zero roots connected.