  It is known that some results for spinors, and in particular for superenergy
spinors, are much less transparent and require a lot more effort to establish,
when considered from the tensor viewpoint. In this paper we demonstrate how the
use of dimensionally dependent tensor identities enables us to derive a number
of 4-dimensional identities by straightforward tensor methods in a signature
independent manner. In particular, we consider the quadratic identity for the
Bel-Robinson tensor ${\cal T}_{abcx}{\cal T}^{abcy} = \delta_x^y {\cal
T}_{abcd}{\cal T}^{abcd}/4$ and also the new conservation laws for the
Chevreton tensor, both of which have been obtained by spinor means; both of
these results are rederived by {\it tensor} means for 4-dimensional spaces of
any signature, using dimensionally dependent identities, and also we are able
to conclude that there are no {\it direct} higher dimensional analogues. In
addition we demonstrate a simple way to show non-existense of such identities
via counter examples; in particular we show that there is no non-trivial Bel
tensor analogue of this simple Bel-Robinson tensor quadratic identity. On the
other hand, as a sample of the power of generalising dimensionally dependent
tensor identities from four to higher dimensions, we show that the symmetry
structure, trace-free and divergence-free nature of the four dimensional
Bel-Robinson tensor does have an analogue for a class of tensors in higher
dimensions.
