= Solution
Apply <energy uniqueness for a linear elastic traction problem>. Subtract two solutions with the same loading, writing their <displacement> difference as $\mathbf v$. Linearity makes its <body force> and <traction>[boundary traction] zero. The preceding energy identity therefore yields $\int_B2W(e(\mathbf v))\,dV=0$. <Positive definiteness> and continuity imply $e(\mathbf v)=0$ everywhere. By the permitted zero-strain characterization,
$$
\boxed{\mathbf v=\mathbf a+\boldsymbol\omega\times\mathbf x.}
$$
Thus the solution is unique modulo infinitesimal rigid <translations> and <rotations>, on each connected body. Conversely adding such a <displacement> changes neither <strain> nor <stress>, so these are exactly the unconstrained modes. This is conditional <uniqueness>; existence of a pure-<traction> solution additionally requires balanced resultant force and <torque>.
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