  We study some properties of the {\bf cd}-index of the Boolean lattice. They
are extremely similar to the properties of the {\ab}-index, or equivalently,
the flag $h$-vector of the Boolean lattice and hence may be viewed as their
{\bf cd}-analogues. We define a different algebra structure on the polynomial
algebra $k < \cv, \dv>$ and give a derivation on this algebra. It is of
significance for the Boolean lattice and forms our main tool. Using similar
methods, we also prove some results for the {\bf cd}-index of the cubical
lattice. We show that the Dehn-Sommerville relations for the flag $f$-vector of
an Eulerian poset are equivalent to certain simple identities that exist in our
algebra.
