The $B$-mode polarization in the cosmic microwave background (CMB) anisotropies at large angular scales provides compelling evidence for the primordial gravitational waves (GWs). It is often stated that a discovery of the GWs establishes the quantum fluctuation of vacuum during the cosmic inflation. Since the GWs could also be generated by source fields, however, we need to check if a sizable signal exists due to such source fields before reaching a firm conclusion when the $B$ mode is discovered. Source fields of particular types can generate non-Gaussianity (NG) in the GWs. Testing statistics of the $B$ mode is a powerful way of detecting such NG. As a concrete example, we show a model in which gauge field sources chiral GWs via a pseudoscalar coupling and forecast the detection significance at the future CMB satellite LiteBIRD. Effects of residual foregrounds and lensing $B$ mode are both taken into account. We find the $B$-mode bispectrum ``BBB'' is in particular sensitive to the source-field NG, which is detectable at LiteBIRD with a $>3\ensuremath{\sigma}$ significance. Therefore the search for the BBB will be indispensable toward unambiguously establishing quantum fluctuation of vacuum when the $B$ mode is discovered. We also introduced the Minkowski functional to detect the NGs. While we find that the Minkowski functional is less efficient than the harmonic-space bispectrum estimator, it still serves as a useful cross-check. Finally, we also discuss the possibility of extracting clean information on parity violation of GWs and new types of parity-violating observables induced by lensing.