  We classify all left-invariant pseudo-Riemannian Einstein metrics on
$\mathrm{SL}(2,\mathbb{R})\times \mathrm{SL}(2,\mathbb{R})$ that are
bi-invariant under a one-parameter subgroup. We find that there are precisely
two such metrics up to homothety, the Killing form and a nearly pseudo-K\"ahler
metric.
