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As a consequence we show the existence of strictly Bach-flat left-invariant metrics on <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$SU(2)\\times \\mathbb {R}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>S<\/mml:mi>\n                    <mml:mi>U<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>\u00d7<\/mml:mo>\n                    <mml:mi>R<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> whose restriction to <jats:italic>SU<\/jats:italic>(2) may be positive definite, Lorentzian or degenerate.\n<\/jats:p>","DOI":"10.1007\/s00332-025-10183-2","type":"journal-article","created":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T13:30:32Z","timestamp":1751635832000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Conformally Einstein Lorentzian Lie Groups: The Non-solvable Case"],"prefix":"10.1007","volume":"35","author":[{"given":"E.","family":"Calvi\u00f1o-Louzao","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"E.","family":"Garc\u00eda-R\u00edo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"I.","family":"Guti\u00e9rrez-Rodr\u00edguez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R.","family":"V\u00e1zquez-Lorenzo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,7,4]]},"reference":[{"key":"10183_CR1","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1016\/j.crma.2013.04.011","volume":"351","author":"E Abbena","year":"2013","unstructured":"Abbena, E., Garbiero, S., Salamon, S.: Bach-flat Lie groups in dimension 4. 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