  The purpose of the present note is to contribute in clarifying the relation
between representation bases used in the closure for the redistribution
(pressure-strain) tensor $\phi_{ij}$, and to construct representation bases
whose elements have clear physical significance. The representation of
different models in the same basis is essential for comparison purposes, and
the definition of the basis by physically meaningfull tensors adds insight to
our understanding of closures. The rate-of-production tensor can be split into
production by mean strain and production by mean rotation $P_{ij}=P_{\bar
S_{ij}}+P_{\bar\Omega_{ij}}$. The classic representation basis
$\mathfrak{B}[\tsr{b}, \tsrbar{S}, \tsrbar{\Omega}]$ of homogeneous turbulence
{\em [{\em eg} Ristorcelli J.R., Lumley J.L., Abid R.: {\it J. Fluid Mech.}
{\bf 292} (1995) 111--152]}, constructed from the anisotropy $\tsr{b}$, the
mean strain-rate $\tsrbar{S}$, and the mean rotation-rate $\tsrbar{\Omega}$
tensors, is interpreted, in the present work, in terms of the relative
contributions of the deviatoric tensors $P^{(\mathrm{dev})}_{\bar
S_{ij}}:=P_{\bar S_{ij}}-\tfrac{2}{3}P_\mathrm{k}\delta_{ij}$ and
$P^{(\mathrm{dev})}_{\bar\Omega_{ij}}:=P_{\bar\Omega_{ij}}$. Different
alternative equivalent representation bases, explicitly using
$P^{(\mathrm{dev})}_{\bar S_{ij}}$ and $P_{\bar\Omega_{ij}}$ are discussed, and
the projection rules between bases are caclulated, using a matrix-based
systematic procedure. An initial term-by-term {\em a priori} investigation of
different second-moment closures is undertaken.
