On some connections between braces and pre-Lie rings outside of the context of Lazard's correspondence

Let $p>3$ be a prime number and let $A$ be a brace whose additive group is a direct sum of cyclic groups of cardinalities larger than $p^{\alpha }$ for some $\alpha$. Suppose that either (i) $A^{\lfloor \frac{p-1}{4}\rfloor }\subseteq pA$ or that (ii) the additive group of brace $A$ has rank smaller than $\lfloor \frac{p-1}{4}\rfloor$. It is shown that for every natural number $i\le \alpha -\frac{4\alpha }{p-1}$ the factor brace $A\mathrm{/}{p}^{i}A$ is obtained by a formula similar to the group of flows from a left nilpotent pre-Lie ring.