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-Kähler 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 Kähler reduction in the above sense is abelian.

25 pages, 1 table. v2: corrected the Lie algebras appearing in Theorem 5.7 and Table 1; added Remark 5.9 concerning isomorphisms between Lie algebras in Table 1; typos corrected; presentation improved