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

Classification of left-invariant Einstein metrics on  $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ that are  bi-invariant under a one-parameter subgroup