We report fully differential cross sections for simultaneous double ionization of atomic beryllium by fast electron impact. We analyze separately the two ionization channels that leave the dication in either the $2{s}^{2}$ autoionizing excited state or its $1{s}^{2}$ bound ground state. For the double continuum, owing to the two slow emitted electrons, we employ the three-pairwise-Coulomb-interaction model (3C) along with the first Born approximation. Aiming to probe the hitherto largely unexplored role of the electron correlation in $(e,3e)$, we determine and check fully correlated, compact analytical wave functions, satisfying all two-particle Kato cusp conditions for the four-electron Be initial state and the two-electron ${\text{Be}}^{2+}$ final states, as well as other, much or little or noncorrelated, functions, issuing from either analytic global optimization or Hartree-Fock theory. We point out similarities or contrasts with trends observed in the double photoionization, allowing for parallels between $(e,3e)$ and $(\ensuremath{\gamma},2e)$. Various numerical examples serve to navigate possible future experiments in the nonrelativistic regime of the ionization problem.