u g (v, w) = g (\nabla_{u} v, w) + g (v, \nabla_{u} w),

and similarly for the two cyclic permutations. Add the identities with leading derivatives

u

and

v

, subtract the one with leading derivative

w

, and use

\nabla_{v} u = \nabla_{u} v - [u, v], \nabla_{u} w = \nabla_{w} u + [u, w], \nabla_{v} w = \nabla_{w} v + [v, w] .

Cancellation and symmetry of

g

give

u (g (v, w)) + v (g (u, w)) - w (g (u, v)) = 2 g (\nabla_{u} v, w) - g ([u, v], w) + g (v, [u, w]) + g (u, [v, w]) .

This is the Koszul formula written with the bracket terms on the right.

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