A simple tensorial proof for the completely symmetric property of the Bel-Robinson tensor

The Bel-Robinson tensor $T_{\alpha \beta \mu \nu }$ was proposed in 1958. The main application of this tensor is for describing gravitational energy. It is known that $T_{\alpha \beta \mu \nu }$ has many nice properties such as being completely symmetric. It is easy to prove this property using spinors as shown in Penrose’s book. The present paper provides an alternative way, using the tensorial method to prove that $T_{\alpha \beta \mu \nu }$ is indeed totally symmetric. Moreover, we also found that the well known identity in vacuum, $R_{\alpha\lambda\sigma\tau}R_{\beta}{}^{\lambda\sigma\tau} \equiv 1/4g_{\alpha\beta}R_{\rho\lambda\sigma\tau} R^{\rho\lambda\sigma\tau}$, which can be proven by the same tensorial method.