We study the trispectrum of the gravitational waves (GWs) generated through the dynamics of an axionic spectator field and SU(2) gauge fields during inflation. In non-Abelian gauge theory, the gauge fields have four-point self-interactions, which induce the tree-level GW trispectrum. We formulate this type of the GW trispectrum including the non-dynamical contributions and evaluate it in the equilateral limit as a unique signal of this model. We find that the ratio of the GW trispectrum to the cube of the scalar power spectrum can be as large as O(106) in the viable parameter space, which could be captured in the CMB observations.
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