Dominant property for the Bel-Robinson tensor and tensor S

The Bel-Robinson tensor contains many nice mathematical properties and its dominant energy condition is desirable for describing the positive gravitational energy. The dominant property is a basic requirement for the quasi-local mass, i.e., in small sphere limit. We claim that there exists another option, a linear combination between the Bel-Robinson tensor $B$ and tensor $S$, which contributes the same dominant property. Moreover, using the 5 Petrov types as the verification, we found that this dominant property justification for the Bel-Robinson tensor can be simplified as examining $B_{0000}\geq|B_{\alpha\beta{}00}|$ and $B_{0000}\geq|B_{0123}|$, instead of $B_{0000}\geq|B_{\alpha\beta\lambda\sigma}|$ for all $\alpha,\beta,\lambda,\sigma=0,1,2,3$.