Invariance of Killing Form

Question

I'm currently learning about Killing Forms and I came across this important property: up to scalar multiples, every simple Lie Algebra has a unique bilinear form that is invariant under all automorphisms of the Lie Algebra.

My question is: why would we want a bilinear form to be invariant? That is, if we act on a Lie Algebra $\mathfrak{g}$ by a linear map, then aren't we changing the geometry of the Lie Algebra (e.g: scaling it in some direction, or skewing it), as most linear maps do? In that case, we wouldn't want the bilinear form to remain invariant under all those changes! My intuition is that most linear maps (on general vector spaces) do not leave inner products invariant, so why would we require that property in this case? Thanks for the help!

Answer 1

One reason is that we obtain a bi-invariant metric. In fact, the following are equivalent:

• The Lie algebra $\mathfrak{g} $ of the Lie group $G$ admits an $\text{Ad}$-invariant inner product,
• The Lie group $G$ admits a bi-invariant metric,

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