  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.
