We introduce in this paper embedded Gaussian unitary ensemble of random matrices, for m fermions in 𝛺 number of single particle orbits, generated by random two-body interactions that are $SU(4)$ scalar, called EGUE$(2)-SU (4)$. Here the $SU (4)$ algebra corresponds to Wigner’s supermultiplet $SU (4)$ symmetry in nuclei. Formulation based on Wigner–Racah algebra of the embedding algebra $U (4\Omega) \supset U (\Omega) \bigotimes SU (4)$ allows for analytical treatment of this ensemble and using this analytical formulas are derived for the covariances in energy centroids and spectral variances. It is found that these covariances increase in magnitude as we go from EGUE(2) to EGUE(2)-s to EGUE(2)-$SU (4)$ implying that symmetries may be responsible for chaos in finite interacting quantum systems.