  We construct indefinite Einstein solvmanifolds that are standard, but not of
pseudo-Iwasawa type. Thus, the underlying Lie algebras take the form
$\mathfrak{g}\rtimes_D\mathbb{R}$, where $\mathfrak{g}$ is a nilpotent Lie
algebra and $D$ is a nonsymmetric derivation. Considering nonsymmetric
derivations has the consequence that $\mathfrak{g}$ is not a nilsoliton, but
satisfies a more general condition.
  Our construction is based on the notion of nondiagonal triple on a nice
diagram. We present an algorithm to classify nondiagonal triples and the
associated Einstein metrics. With the use of a computer, we obtain all
solutions up to dimension $5$, and all solutions in dimension $\leq9$ that
satisfy an additional technical restriction.
  By comparing curvatures, we show that the Einstein solvmanifolds of dimension
$\leq 5$ that we obtain by our construction are not isometric to a standard
extension of a nilsoliton.
