Bi-invariant Riemannian metric

A Riemannian metric on a Lie group is bi-invariant if both left and right translations are isometries. Its identity inner product is invariant under the Adjoint representation of a Lie group; infinitesimally,This skew-adjointness is the key additional property beyond left invariance.

Integral curves through the identity of left-invariant vector fields are one-parameter subgroups, by flow uniqueness and left translation. They extend for all time because a fixed local existence interval translates to every point. Under a bi-invariant Riemannian metric they are geodesics by the Levi-Civita connection of a bi-invariant metric. Geodesic uniqueness proves that these are all geodesics through the identity.

For left-invariant vector fields, the Koszul formula and the adjoint invariance of a bi-invariant Riemannian metric give the displayed Levi-Civita connection. In particular ${\nabla }_{X}X=0$.