  The negative cyclic homology for a differential graded algebra over the
rational field has a quotient of the Hochschild homology as a direct summand if
the $S$-action is trivial. With this fact, we show that the string bracket in
the sense of Chas and Sullivan is reduced to the loop product followed by the
BV operator on the loop homology provided the given manifold is BV exact. The
reduction is indeed derived from the equivalence between the BV exactness and
the triviality of the $S$-action. Moreover, it is proved that a Lie bracket on
the loop cohomology of the classifying space of a connected compact Lie group
possesses the same reduction. By using these results, we consider the
non-triviality of string brackets. Another highlight is that a simply-connected
space with positive weights is BV exact. Furthermore, the higher BV exactness
is also discussed featuring the cobar-type Eilenberg-Moore spectral sequence.
