Bi-invariant metrics on compact Lie groups

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$ .

Theorem. $G$ admits a bi-invariant Riemannian metric. Equivalently, there exists an inner product on $\mathfrak{g}$ invariant under the adjoint action of $G$ .

Remarks

Idea of construction (averaging). Start with any inner product $\langle\cdot,\cdot\rangle_0$ on $\mathfrak{g}$ , and define a new inner product by averaging over $G$ using Haar measure:

\langle X,Y\rangle := \int_G \langle \mathrm{Ad}_g X,\, \mathrm{Ad}_g Y\rangle_0 \, dg.

This averaged form is $\mathrm{Ad}(G)$ -invariant by construction, and it induces a bi-invariant metric on $G$ via left translation.

Consequences. With a bi-invariant metric, geodesics through the identity are exactly one-parameter subgroups, making the exponential map geometrically canonical. This also provides natural bi-invariant volume forms and simplifies curvature computations.