  We study a class of left-invariant pseudo-Riemannian Sasaki metrics on
solvable Lie groups, which can be characterized by the property that the zero
level set of the moment map relative to the action of some one-parameter
subgroup $\{\exp tX\}$ is a normal nilpotent subgroup commuting with $\{\exp
tX\}$, and $X$ is not lightlike. We characterize this geometry in terms of the
Sasaki reduction and its pseudo-K\"ahler quotient under the action generated by
the Reeb vector field.
  We classify pseudo-Riemannian Sasaki solvmanifolds of this type in dimension
$5$ and those of dimension $7$ whose K\"ahler reduction in the above sense is
abelian.
