The electronic structure of ErGa3 and its isostructural compounds with Tm, Yb, and Lu are investigated with a highly accurate band structure scheme in LDA and GGA and warped muffin-tin approximation. In contrast to other investigations, the 4f electrons of the constituent Re are also treated as part of the valence bands. The position of the corresponding 4f bands relative to the Fermi energy E-F strongly depends on the nuclear charge of Re. In Lu, they lie almost by 0.5 Ryd below E-F and are extremely narrow. In Er, both in LDA and GGA, the 4f bands are found to be very close to the Fermi level E-F. Assuming most of the 4f electrons to be part of the core removes the disagreement almost completely but produces a Fermi surface with a topology markedly different from that proposed in previous investigations. The intersections of the Fermi surface with planes are strongly varying within the Brillouin zone, they do not well match with the sparse experimental results. Investigations using the LDA + U scheme as well as investigations of the dielectric response function are sketched.