Exponent p implies associated Lie ring is (p-1)-Engel Lie algebra over field of p elements

Statement
Suppose $$p$$ is a prime number. Suppose $$G$$ is a group of prime exponent with exponent $$p$$ and $$L(G)$$ is its associated Lie ring. Then, the following two thing are true for $$L(G)$$:


 * 1) The additive group of $$L(G)$$ is also a group of prime exponent, specifically exponent $$p$$. Thus, $$L(G)$$ is a Lie algebra over the field of $$p$$ elements.
 * 2) $$L(G)$$ is a $$(p - 1)$$-Engel Lie ring. In other words:

$$[x,[x,\dots,[x,y]\dots]] = 0$$

for all $$x,y \in L$$ where $$x$$ is written $$p - 1$$ times.