Kreknin's theorem on automorphism of finite order of Lie ring

Statement
Suppose $$L$$ is a Lie ring and $$n$$ is a natural number. Suppose $$\varphi$$ is an automorphism of $$L$$ of order $$n$$. Define:

$$nL := \{ nx \mid x \in L \}$$

where $$nx$$ is $$x$$ added to itself $$n$$ times.

Denote by $$k(n)$$ the value of Kreknin's function of $$n$$. Denote by $$(nL)^{(k(n))}$$ the $$k(n)^{th}$$ member of the derived series of $$nL$$. Denote by $$C_L(\varphi)$$ the set of fixed points in $$L$$ under $$\varphi$$.

Then, $$(nL)^{k(n)}$$ is contained in the ideal generated by $$C_L(\varphi)$$.

Related facts

 * Kreknin's theorem on existence of Kreknin's function
 * Higman's theorem on automorphism of prime order of Lie ring