Exterior derivative of contracted volume form
= Exterior derivative of contracted volume form
{title2=$d(\iota_X\omega_g)=(\operatorname{div}X)\omega_g$}
At a point in a <normal orthonormal frame>, torsion-freeness makes the frame brackets vanish there, hence the dual one-forms have zero <exterior derivative> there. Differentiating the <interior product> formula then leaves only $\sum_iE_i(f_i)\omega_g$, the <divergence of a Riemannian vector field>. Since the point is arbitrary, the identity holds globally.