  The condensation of spin-orbit-induced excitons in $(t_{2g})^4$ electronic
systems is attracting considerable attention. In the large Hubbard U limit,
antiferromagnetism was proposed to emerge from the Bose-Einstein Condensation
(BEC) of triplons ($J_{\textrm{eff}} = 1$). In this publication, we show that
even for the weak and intermediate U regimes, the spin-orbit exciton
condensation is possible leading also to staggered magnetic order. The
canonical electron-hole excitations (excitons) transform into local triplon
excitations at large U , and this BEC strong coupling regime is smoothly
connected to the intermediate U excitonic insulator region. We solved the
degenerate three-orbital Hubbard model with spin-orbit coupling ($\lambda$) in
one-dimensional geometry using the Density Matrix Renormalization Group, while
in two-dimensional square clusters we use the Hartree-Fock approximation (HFA).
Employing these techniques, we provide the full $\lambda$ vs U phase diagrams
for both one- and two- dimensional lattices. Our main result is that at the
intermediate Hubbard U region of our focus, increasing $\lambda$ at fixed U the
system transitions from an incommensurate spin-density-wave metal to a
Bardeen-Cooper-Schrieffer (BCS) excitonic insulator, with coherence length r
coh of O(a) and O(10a) in 1d and 2d, respectively, with a the lattice spacing.
Further increasing $\lambda$, the system eventually crosses over to the BEC
limit (with r coh << a).
