Exercise 2, § 4

Tag 00WD, permanent. Cite it as [Bourbaki, Tag 00WD].

Lie Groups and Lie Algebras, VII, 63

Construct a Lie group G such that the function $r^0_{Ad}$ is not constant. (Take $\mathfrak{g}$ abelian $\not= 0$ and Ad non-trivial.)