  We give a new description of the flag major index, introduced by Adin and
Roichman, by using a major index defined by Reiner. This allows us to establish
a connection between an identity of Reiner and some more recent results due to
Chow and Gessel. Furthermore we generalize the main identity of Chow and Gessel
by computing the four-variate generating series of descents, major index,
length, and number of negative entries over Coxeter groups of type $B$ and $D$.
