  In this note, we discuss symmetric brackets on skew-symmetric algebroids
associated with a metric structure. Given a pseudo-Riemannian metric structure,
we describe symmetric brackets induced by connections with totally
skew-symmetric torsion in the language of Lie derivatives and differentials of
functions. In particular, we obtain an explicit formula of the Levi-Civita
connection. We also present some symmetric brackets on almost Hermitian
manifolds. Especially, we discuss the first canonical Hermitian connection and
show the formula for it in the case of nearly Kaehler manifolds using the
properties of symmetric brackets.
