  Cerium-hexaboride (CeB$_6$) f-electron compound displays a rich array of
low-temperature magnetic phenomena, including `magnetically hidden' order,
identified as multipolar in origin via advanced x-ray scattering. From
first-principles electronic-structure results, we find that the
\emph{antiferro-quadrupolar} (AFQ) ordering in CeB$_{6}$ arises from
crystal-field splitting and yields band structure in agreement with
experiments. With interactions of $p$-electrons between Ce and B$_{6}$ being
small, the electronic state of CeB$_{6}$ is suitably described as
Ce(4$f^{1}$)$^{3+}$(e$^{-}$)(B$_{6}$)$^{2-}$. The AFQ state of orbital spins is
caused by an exchange interaction induced through spin-orbit interaction, which
also splits J=5/2 state into $\Gamma_{8}$ ground state and $\Gamma_{7}$ excited
state. Within the smallest antiferromagnetic (111) configuration, an
orbital-ordered AFQ state appears during charge self-consistency, and supports
the appearance of `hidden' order. Hydrostatic pressure (either applied or
chemically induced) stabilizes the AFM (AFQ) states over a ferromagnetic one,
as observed at low temperatures.
