| Summary: | In this paper, we study left-invariant Einstein-like metrics on the compact Lie group <i>G</i>. Assume that there exist two subgroups, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>⊂</mo><mi>K</mi><mo>⊂</mo><mi>G</mi></mrow></semantics></math></inline-formula>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>/</mo><mi>K</mi></mrow></semantics></math></inline-formula> is a compact, connected, irreducible, symmetric space, and the isotropy representation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>/</mo><mi>H</mi></mrow></semantics></math></inline-formula> has exactly two inequivalent, irreducible summands. We prove that the left metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>⟨</mo><mo>·</mo><mo>,</mo><mo>·</mo><mo>⟩</mo></mrow><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub></mrow></msub></semantics></math></inline-formula> on <i>G</i> defined by the first equation, must be an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>-metric. Moreover, we prove that compact Lie groups do not admit non-naturally reductive left-invariant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">B</mi></semantics></math></inline-formula>-metrics, such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>⟨</mo><mo>·</mo><mo>,</mo><mo>·</mo><mo>⟩</mo></mrow><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub></mrow></msub></semantics></math></inline-formula>.
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