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For a left-invariant metric on a Lie group, the one-parameter subgroup ${\gamma }_{\xi }(t)=\exp (t\xi )$ is a geodesic exactly whenfor every $\eta$ in the Lie algebra. The Koszul formula gives $⟨{\nabla }_{\xi }\xi ,\eta ⟩=-⟨\xi ,[\xi ,\eta ]⟩$.

The Levi-Civita connection is also a metric connection, soand similarly for the two cyclic permutations. Add the identities with leading derivatives $u$ and $v$, subtract the one with leading derivative $w$, and useCancellation and symmetry of $g$ giveThis is the Koszul formula written with the bracket terms on the right.