Higman's theorem on automorphism of prime order of Lie ring

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

$$pL := \{ px \mid x \in L \}$$

where $$px$$ is $$x$$ added to itself $$p$$ times.

Denote by $$h(p)$$ the value of Higman's function of $$p$$. Denote by $$\gamma_{h(p) + 1}(pL)$$ the $$(h(p) + 1)^{th}$$ member of the lower central series of $$pL$$.

Then, $$\gamma_{h(p) + 1}(pL)$$ is contained in the ideal of $$L$$generated by $$C_L(\varphi)$$.

Related facts

 * Higman's theorem on existence of Higman's function
 * Kreknin's theorem on automorphism of finite order of Lie ring: This works for automorphisms of any finite order, but uses the derived series instead of the lower central series, and uses Kreknin's function instead of Higman's function.