  Algebraic conditions on the Ricci tensor in the Rainich-Misner-Wheeler
unified field theory are known as the Rainich conditions. Penrose and more
recently Bergqvist and Lankinen made an analogy from the Ricci tensor to the
Bel-Robinson tensor $B_{\alpha\beta\mu\nu}$, a certain fourth rank tensor
quadratic in the Weyl curvature, which also satisfies algebraic Rainich-like
conditions. However, we found that not only does the tensor
$B_{\alpha\beta\mu\nu}$ fulfill these conditions, but so also does our recently
proposed tensor $V_{\alpha\beta\mu\nu}$, which has many of the desirable
properties of $B_{\alpha\beta\mu\nu}$. For the quasilocal small sphere limit
restriction, we found that there are only two fourth rank tensors
$B_{\alpha\beta\mu\nu}$ and $V_{\alpha\beta\mu\nu}$ which form a basis for good
energy expressions. Both of them have the completely trace free and causal
properties, these two form necessary and sufficient conditions. Surprisingly
either completely traceless or causal is enough to fulfill the algebraic
Rainich conditions. Furthermore, relaxing the quasilocal restriction and
considering the general fourth rank tensor, we found two remarkable results:
(i) without any symmetry requirement, the algebraic Rainich conditions only
require totally trace free; (ii) with a symmetry requirement, we recovered the
same result as in the quasilocal small sphere limit.
