Fraction of tuples for iterated Lie bracket word in subring of finite Lie ring is at least as much as in whole ring

Statement
Suppose $$w$$ is a Lie ring word in $$n$$ letters that involves iterations of the Lie bracket only, and where every letter appears exactly once. For a finite Lie ring $$R$$, consider the fraction:

$$f_w(R) := \frac{|\{(a_1,a_2,\dots,a_n) \in R^n \mid w(a_1,a_2,\dots,a_n) = 0\}|}{|R|^n}$$

Then, if $$S$$ is a subring of $$R$$, we have:

$$f_w(S) \ge f_w(R)$$

The possibilities for $$w$$ include words such as $$\dots[[x_1,x_2],x_3],\dots,x_{n-1}],x_n]$ (which define nilpotency class $\le n-1$ and words such as $[[x_1,x_2],[x_3,x_4$$ (which defines derived length $$2$$).

Facts used

 * 1) uses::Fraction of tuples satisfying groupy relation in subgroup is at least as much as in whole group

Related facts

 * Commuting fraction in subring of finite non-associative ring is at least as much as in whole ring
 * Commuting fraction in subring of finite Lie ring is at least as much as in whole ring
 * Associating fraction in subring of finite non-associative ring is at least as much as in whole ring
 * Commuting fraction in subgroup is at least as much as in whole group