For two interacting spinor fields with equal masses and the use of relativistically invariant gauges of Lorentz type, a quasipotential is constructed in accordance with the first variant of the Logunov-Tavkhelidze potential approach. The diagrams of second order are taken into account, and the quasipotential is found in the four-component Dirac form and in the two-component Pauli form. The low-energy bound of the two-component representation is found. In the case of the Landau and Fried-Yennie gauges the recovery of the complete structure of the Breit potential is demonstrated.