  We reformulate the Shiromizu, Maeda and Sasaki (SMS) braneworlds within the
framework of the five-dimensional Einstein equations. In many applications of
the braneworld Einstein field equations, the Weyl term is attributed to the
bulk, thus splitting the non-Einsteinian terms into `bulk' and `brane' terms.
Here by employing standard geometrical identities, we show that such a split is
non-unique, since these terms get mixed up in different formulations. An
important consequence of this non-uniqueness is that even though the full
brane-bulk systems in all such formulations are completely equivalent,
important differences can arise were one to truncate different formulations by
throwing away the associated `bulk' terms. This is particularly likely to be
the case in more general anisotropic/inhomogeneous settings with non-AdS bulks,
in which the usual truncation of the SMS (which throws away the Weyl term)
would not coincide with the full system. We emphasize that rather than
providing support for any truncation, these differences show clearly the
dangers of using any truncated equations and provide a strong argument in
favour of studying the full brane-bulk system. The different formulations we
provide also permit different ways of approaching the full brane-bulk system
which may greatly facilitate its study. An example of this is the second-order
nature of the formulations given here as opposed to the SMS's formulation which
is third-order.
