Which Lie groups have Lie algebras admitting an Ad-invariant inner product?

Solution 1:

The answer can be found in the reference given by muzzlator. Combining proposition 18.3 on page 511 and theorem 18.8 on page 514, we see that the following are equivalent:

• The Lie algebra of the Lie group $G$ admits an $\text{Ad}$-invariant inner product,
• The Lie group $G$ admits a bi-invariant metric,
• The Lie group $G$ is isomorphic to the cartesian product of a compact group and a vector space $\mathbb{R}^n$.

The equivalence between the first two statements is relatively straightforward, and a proof is given in the reference linked to above. The equivalence between the last two statements is stated in the above reference, but not proved (though the proof is sketched). One refers to lemma 7.5 of Milnor's paper Curvatures of Left Invariant Metrics on Lie Groups (1976) for a proof.