  The effective electron-electron interaction in the electron gas depends on
both the density and spin local field factors. Variational Diagrammatic Quantum
Monte Carlo calculations of the spin local field factor are reported and used
to quantitatively present the full spin-dependent, electron-electron
interaction. Together with the charge local field factor from previous
Diffusion Quantum Monte Carlo calculations, we obtain the complete form of the
effective electron-electron interaction in the uniform three-dimensional
electron gas. Very simple quadratic formulas are presented for the local field
factors that quantitatively produce all of the response functions of the
electron gas at metallic densities.
  Exchange and correlation become increasingly important at low densities. At
the compressibility divergence at rs = 5.25, both the direct (screened Coulomb)
term and the charge-dependent exchange term in the electron-electron
interaction at q=0 are separately divergent. However, due to large
cancellations, their difference is finite, well behaved, and much smaller than
either term separately. As a result, the spin contribution to the
electron-electron interaction becomes an important factor. The static
electron-electron interaction is repulsive as a function of density but is less
repulsive for electrons with parallel spins.
  The effect of allowing a deformable, rather than rigid, positive background
is shown to be as quantitatively important as exchange and correlation. As a
simple concrete example, the electron-electron interaction is calculated using
the measured bulk modulus of the alkali metals with a linear phonon dispersion.
The net electron-electron interaction in lithium is attractive for wave vectors
$0-2k_F$, which suggests superconductivity, and is mostly repulsive for the
other alkali metals.
