= Adjoint dilation obstruction to a bi-invariant Riemannian metric
If a <Lie group> has an <Adjoint representation of a Lie group> element with a real eigenvector of eigenvalue whose modulus differs from one, it has no <bi-invariant Riemannian metric>. Conjugation is a composition of left and right translations, hence would preserve the positive identity inner product. Applying it to that eigenvector gives $\|v\|=|\lambda|\|v\|$, a contradiction. The <orientation-preserving affine group of the real line> supplies a connected example: conjugation by dilation of factor two multiplies the infinitesimal translation by two.
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