Relativistic density functional theory (DFT) computations were performed to investigate the dynamic motion of an encapsulated Lu pair inside a C₇₆(T_d) cage. The results revealed that the lowest-energy configuration of Lu₂@C₇₆(T_d) adopts C₂ symmetry; four electrons are transferred to the outer carbon cage and the two encapsulated Lu atoms form a metal–metal single bond (with an electronic structure of Lu₂⁴⁺@C₇₆⁴⁻), and the good electron delocalization in the C₇₆⁴⁻(T_d) cage partially contributes the thermodynamic preference of Lu₂@C₇₆(T_d). The rather small barrier (3.2 kcal mol⁻¹) for Lu₂ atoms to hop from one stable site to another leads to flexible motion of the Lu pair inside the parent fullerene cage, and the O_h symmetrical motion trajectory of two Lu atoms is consistent with the STM image. The computed ¹³C NMR spectrum with this trajectory also agrees well with the experimental results.