Commuting fraction in subring of finite Lie ring is at least as much as in whole ring

Statement
Suppose $$R$$ is a finite fact about::Lie ring (i.e., $$R$$ is a Lie ring whose underlying set is finite). Suppose $$S$$ is a subring of $$R$$. Then, the commuting fraction of $$S$$ is at least as much as that of $$R$$.

In symbols, if $$CP(R) := \{ (x,y) \in R^2 \mid [x,y] = 0 \}$$ and $$CP(S) = S^2 \cap CP(R)$$, then:

$$\frac{|CP(S)|}{|S|^2} \ge \frac{|CP(R)|}{|R|^2}$$

In fact, the result also holds if $$S$$ is simply an additive subgroup of $$R$$ and not a subring.

Facts used

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

Related facts

 * Fraction of tuples for iterated Lie bracket word in subring of finite Lie ring is at least as much as in whole ring
 * Commuting fraction in subring of finite non-associative ring is at least as much as in whole ring
 * Associating fraction in subring of 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

Proof
The proof uses fact (1), and the observation that the relation of commuting in a Lie ring is a groupy relation.