  We provide two new formulations of the separativity problem.
  First, it is known that separativity (and strong separativity) in von Neumann
regular (and exchange) rings is tightly connected to unit-regularity of certain
kinds of elements. By refining this information, we characterize separative
regular rings in terms of a special type of inner inverse operation, which is
defined via a single identity. This shows that the separative regular rings
form a subvariety of the regular rings. The separativity problem reduces to the
question of whether every element of the form $(1-aa')bac(1-a'a)$ in a regular
ring is unit-regular, where $a'$ is an inner inverse for $a$.
  Second, it is known that separativity in exchange rings is equivalent to
regular matrices over corner rings being reducible to diagonal matrices via
elementary row and column operations. We show that for $2\times 2$ invertible
matrices, four elementary operations are sufficient, and in general also
necessary. Dropping the invertibility hypothesis, but specializing to
separative regular rings, we show that three elementary row operations together
with three elementary column operations are sufficient, and again in general
also necessary. The separativity problem can subsequently be reframed in terms
of the explicit number of operations needed to diagonally reduce.
