in other words, the [**ab**]{}-index) of the Boolean lattice displays many remarkable patterns. These have been studied in detail in . The goal of this paper is to establish the [**cd**]{}-analogues of these properties for the Boolean lattice.

The intuitive reason why properties of the ${{{\bf a}}{{\bf b}}}$-index carry over to the ${{{\bf c}}{{\bf d}}}$-index is explained in Appendix \[s:connection\]. Unfortunately, the methods used to study the two problems are totally different. The coefficients of the ${{{\bf a}}{{\bf b}}}$-index of the Boolean lattice are related to the descent statistic of permutations and they can be studied effectively using that interpretation . Though there are similar interpretations for the coefficients of the ${{{\bf c}}{{\bf d}}}$-index (see item (1) in Section \[ss:questions\]), we do not know how to use them. Instead, our method is based on an algebraic study of the polynomial algebra ${\mathcal F} = k \langle {{\bf c}},{{\bf d}}\rangle$ in the non-commuting variables [**c**]{} and [**d**]{}.

The [**cd**]{}-index of the Boolean lattice for small ranks is shown in Figure \[f:bl\]. The letter ${e}$ is a formal symbol that is added to ${{\mathcal F}}$ in degree $-1$, see Section \[ss:ext\].

Questions (and partial answers) {#ss:questions}
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For $v$, a [**cd**]{}-monomial of 