Bound on prime power divisors in associative formula equals bound on prime power divisors in Lie formula for Lie element

Statement
The gist of the statement is that if a formula for a Lie element exists in the associative algebra, then:


 * 1) There exists a (unique) formula in terms of basic Lie products such that for any prime $$p$$, the largest power of $$p$$ dividing any denominator in that formula is the same as for the associative formula.
 * 2) There exists a formula in terms of right-normed Lie products (or equivalently, left-normed Lie products) such that for any prime $$p$$, the largest power of $$p$$ dividing any denominator in that formula is the same as for the associative formula.