Multipole order parameters with spin-orbit coupling in the $t_{2g}$ subspace are systematically investigated based on symmetry analysis and tensor-coupling methods. Through a classification according to their space-time symmetry, ranks, as well as point groups ($O_h$ and $D_{3d}$), the distinct physical significance of these multipoles is clarified. Subsequently, their induced parameters under external vector fields are derived, which uncover the various novel physical effects such as magnetodielectric and perpendicular polarization responses, serving as fingerprints for identifying the underlying multipole orders. These results not only resolve ambiguities in the physical significance of $t_{2g}$ multipoles, but also provide a concrete roadmap for their experimental detection and manipulation through external field responses.

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• Electrical properties
• Magnetism
• Spin-orbit coupling
• Strongly correlated systems
• Symmetries