  In this manuscript, we investigate a Cartan calculus on the homology of free
loop spaces which is introduced by Kuribayashi, Wakatsuki, Yamaguchi and the
author. In particular, it is proved that the Cartan calculus can be described
by the loop product and bracket in string topology. Moreover, by using the
descriptions, we show that the loop product behaves well with respect to the
Hodge decomposition of the homology of free loop spaces.
