Classification of left-invariant Einstein metrics on SL(2,R)×SL(2,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{SL}(2,\mathbb {R})\times \textrm{SL}(2,\mathbb {R})$$\end{document} that are bi-invariant under a one-parameter subgroup

We classify all left-invariant pseudo-Riemannian Einstein metrics on SL(2,R)×SL(2,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{SL}(2,\mathbb {R})\times \textrm{SL}(2,\mathbb {R})$$\end{document} 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-Kähler metric.