Bound on prime power divisors of denominators in formula for group commutator in terms of Lie bracket

Statement
For a prime $$p$$ and a natural number $$n$$, denote by $$g(p,n)$$ the largest $$k$$ such that, if we truncate the formula for group commutator in terms of Lie bracket to a group of class at most $$n$$ (or equivalently, we look only at the terms of degree at most $$n$$ in the formula), then one or more of the denominators is divisible by $$p^k$$. Then:


 * For all $$n$$ we have:

$$g(p,n) \le \left[\frac{n - 1}{p - 1}\right]$$


 * Further, in the case that $$n \le p$$, we have:

$$g(p,n) = 0$$

This provides a stricter bound than the preceding inequality only in the case that $$n = p$$.

Facts used

 * 1) uses::Bound on prime power divisors of denominators in Baker-Campbell-Hausdorff formula