BCH convergence on a powerful Lie lattice ($x\ast y=\mathrm{BCH}(x,y)$)

The Baker--Campbell--Hausdorff formula converges integrally on a powerful Lie lattice. A degree-$m$ Lie term has bracket divisibility $p^{ϵ(m-1)}$; the standard integral coefficient bound loses at most $\lfloor (m-1)/(p-1)\rfloor$ powers of $p$. The remaining valuation tends to infinity. Associativity follows from the formal exponential identity, the identity is zero, the inverse is $-x$, and $x^{\ast n}=nx$. The resulting pro-p group is a uniform pro-p group, with power subgroups $p^{n}L$.