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Nuclear magnetic resonance (NMR) was recently shown to measure the bulk band inversion of Bi_{2}Se_{3} through changes in the ^{209}Bi nuclear quadrupole interaction, and the corresponding tensor of the local electric field gradient was found to follow, surprisingly, the direction of the external magnetic field if the sample is rotated. This manifests a hidden property of the charge carriers in the bulk of this topological insulator, which is explored here with another material, Bi_{2}Te_{3}. It is found that two electric field gradients appear to be present at ^{209}Bi, one rests with the lattice, as usual, while a second follows the external field if it is rotated with respect to the crystal axes. These electronic degrees of freedom and their lifetime is believed to be responsible for an alternate quadrupolar relaxation that should lead to other special properties including the electronic specific heat.