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The Bel-Robinson tensor $T_{\alpha\beta\mu\nu}$ was proposed in 1958. Themain application of this tensor is for describing gravitational energy. It isknown that $T_{\alpha\beta\mu\nu}$ has many nice properties such as beingcompletely symmetric. It is easy to prove this property using spinors as shownin Penrose's book. The present paper provides an alternative way, using thetensorial method to prove that $T_{\alpha\beta\mu\nu}$ is indeed totallysymmetric. Moreover, we also found that the well known identity in vacuum,$R_{\alpha\lambda\sigma\tau}R_{\beta}{}^{\lambda\sigma\tau} \equiv1/4g_{\alpha\beta}R_{\rho\lambda\sigma\tau} R^{\rho\lambda\sigma\tau}$, whichcan be proven by the same tensorial method.

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