 Solution

Use the faithful logarithm realization from the root Solution: $\log :G\to A$ identifies $L_G$ with the Lie algebra $\log G$. For $x,y\in G$, the convergent exponential identity in $A$ givesEvery Lie polynomial and its convergent sum are preserved by this realization of $L_G$. Therefore the BCH product on the underlying set of $L_G$ has exactly the same logarithm as the original product $xy$. Injectivity of the logarithm proves equality of the two products, with the same topology, identity and inverses:This proves the reverse direction of the uniform pro-p group and powerful Lie lattice correspondence directly; the forward identity alone would not imply it.