($=\uparrow, \downarrow$) are introduced to represent the state $|\tau,\sigma\rangle$, which distinguish non-Kramers’ and Kramers’ pairs, respectively.

Triple-${\textbf k}$ structure
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The four-fold degeneracy in the ground $\Gamma_8^{(2)}$ quartet may be lifted by the intersite interaction, giving rise to induced multipole moments. Actually, several experiments tell us that the time-reversal symmetry is broken with nearly zero dipole moment in the ordered phase below $T_0=25.5$ K.[@Kopmann98; @Tokunaga05] These observations lead Santini and Amoretti to propose the antiferro ordering of $T_{xyz}$-type ($T_{xyz} \equiv \frac{\sqrt{15}}{6}\overline{J_xJ_yJ_z}$). [@Santini00; @Santini02] Here the overline on operators means symmetrization, for example, $\overline{J_{x}J_{y}^2}=J_x J_y^2 + J_y J_x J_y + J_y^2 J_x$.[@Shiina97]. Unfortunately, this phase would not give rise to the RXS intensities observed in the experiments.

An important observation is that no external distortion from cubic structure exists in the ordered phase, that is, the unit cell remains cubic below $T_0$. This leads us to consider the triple-**k** ordering, since it allows the crystal to keep the cubic symmetry. As schematically shown in Fig. \[fig.struc2\] (c), the triple-**k** structure is defined by all three members of the star of ${\textbf k}=\langle001 \rangle$ simultaneously present on each site of the lattice; there are four sublattices 1, 2, 3 and 4 at 