The standard two-step perturbative approach of the gyrokinetic reduction usually consists in a guiding-center reduction and a subsequent gyrocenter reduction. At each step, a Lie transform procedure is applied aiming to eliminate the fast gyroangle dependencies from the one-form of the particle motion. Here we present an alternative two-step procedure: Using a translation in velocity, we embed the Lie transform performed on the symplectic part of the gyrocenter reduction in the guiding-center one. Using a canonical Lie transform, we then eliminate the gyroangle dependence from the Hamiltonian. As a result we present a systematic and self-contained derivation of the second order fully electromagnetic gyrokinetic equations for the code ordering, assuming that the perturbations related to the curvature of the background magnetic field are of higher order with respect to the relative amplitudes of the fluctuating fields.