  Generic rotationally invariant random matrix models satisfy a simple
relation: the probability distribution of off-diagonal elements and the one of
half the difference between any two diagonal elements coincide. In the spirit
of the Eigenstate Thermalization Hypothesis (ETH), we test the hypothesis that
the same relation holds in quantum systems that are non-localized, when one
considers small energy differences. The relation provides a stringent test of
ETH beyond the Gaussian ensemble. We apply it to a disordered spin chain, the
SYK model and a Floquet system.
