  We introduce a gyro-gauge independent formulation of a simplified
guiding-center reduction, which removes the fast time-scale from particle
dynamics by Lie-transforming the velocity vector field. This is close to
Krylov-Bogoliubov method of averaging the equations of motion, although more
geometric. At leading order, the Lie-transform consists in the generator of
Larmor gyration, which can be explicitly inverted, while working with
gauge-independent coordinates and operators, by using the physical gyro-angle
as a (constrained) coordinate. This brings both the change of coordinates and
the reduced dynamics of the minimal guiding-center reduction order by order in
a Larmor radius expansion. The procedure is algorithmic and the reduction is
systematically derived up to full second order, in a more straightforward way
than when Lie-transforming the phase-space Lagrangian or averaging the
equations of motion. The results write up some structures in the guiding-center
expansion. Extensions and limitations of the method are considered.
