We study $\eta$-Einstein Sasakian structures on Lie algebras, that is,
Sasakian structures whose associated Ricci tensor satisfies an Einstein-like
condition. We divide into the cases in which the Lie algebra’s centre is
non-trivial (and necessarily one-dimensional) and those where it is zero. In
the former case we show that any Sasakian structure on a unimodular Lie algebra
is $\eta$-Einstein. As for centreless Sasakian Lie algebras, we devise a
complete characterisation under certain dimensional assumptions regarding the
action of the Reeb vector. Using this result, together with the theory of
normal $j$-algebras and modifications of Hermitian Lie algebras, we construct
new examples of $\eta$-Einstein Sasakian Lie algebras and solvmanifolds, and
provide effective restrictions for their existence.

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