  In an arbitrary complete differential graded Lie algebra, we construct a
group operation $\bullet$ on $L_1$ such that the differential of the product of
two elements is the Baker-Campbell-Hausdorff product of their differentials,
i.e., $d(x\bullet y)=dx\ast dy$. We study some properties of this new structure
and some applications, especially in homotopy theory, where this operation can
be used to construct a Lie model for the 4-simplex. In particular, this solves,
in dimension 4, a problem proposed by Lawrence and Sullivan.
