Commutator subalgebra of the Lie algebra associated with a right-angled Coxeter group

We study the graded Lie algebra $L(RC_K)$ associated with the lower central series of a right-angled Coxeter group $RC_K$. We prove that its commutator subalgebra is a quotient of the polynomial ring over an auxiliary Lie subalgebra $N_K$ of the graph Lie algebra $L_K$, and conjecture that the quotient map is an isomorphism. The epimorphism is defined in terms of a new operation in the associated Lie algebra, which corresponds to the squaring and has an analogue in homotopy theory.