Levi-Civita connection of a bi-invariant metric (${\nabla }_{X}Y=\frac{1}{2}[X,Y]$)

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 ${\nabla }_{X}X=0$.