Is there a relationship between the mass of a SMBH and the kinetic energy of its host elliptical galaxy?

We use a restricted sample of elliptical galaxies, whose kinematical parameters inside the semimajor axis were calculated correcting the effect of the integration of the light along the line of sight, in order to analyze a possible relationship between the mass of a Supermassive Black Hole (SMBH) and the kinetic energy of random motions in the host galaxy. We find $M_{BH} \propto (M_{G} \sigma^{2})^\alpha$ with $0.87 \leq \alpha \leq 1$ depending on the different fitting methods and samples used. This result could be interpreted as a new fundamental relationship or as a new way to explain the old $M_{BH} - \sigma$ law. In fact, the relations of the velocity dispersion both with the mass of the SMBH ($M_{BH} \propto \sigma^{4.12}$) and with the mass of the host galaxy ($M_{G} \propto \sigma^{2.16}$) induce us to infer an almost direct proportionality: $M_{BH} \propto M_{G} \sigma^{2}$. A similar relationship is found for the total kinetic energy involving the rotation velocity too.