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In this note, we discuss symmetric brackets on skew-symmetric algebroidsassociated with a metric structure. Given a pseudo-Riemannian metric structure,we describe symmetric brackets induced by connections with totallyskew-symmetric torsion in the language of Lie derivatives and differentials offunctions. In particular, we obtain an explicit formula of the Levi-Civitaconnection. We also present some symmetric brackets on almost Hermitianmanifolds. Especially, we discuss the first canonical Hermitian connection andshow the formula for it in the case of nearly Kaehler manifolds using theproperties of symmetric brackets.

Symmetric brackets induced by connections with totally skew-symmetric torsion on skew-symmetric algebroids