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 $G$.

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:

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.