Higgins' lemma on Engel conditions

Statement
Suppose $$L$$ is a $$(n+1)$$-Engel Lie ring. Then, the following identities hold for right-normed Lie products assuming that the torsion-free threshold of $$L$$ is at least $$n$$:


 * 1) $$\sum_{i=0}^n x^iyx^{n-i}z = 0 \ \forall x,y,z\in L$$
 * 2) $$\sum_{j=0}^n (-1)^{j+1} \binom{n + 1}{j} x^jyx^{n-j}z = 0 \ \forall x,y,z \in L$$

Shifting to an associative ring
Consider the ring of additive group endomorphisms of $$L$$. This is an associative ring with multiplication defined by composition, hence also can be made into a Lie ring. The Lie ring of inner derivations, given by the adjoint actions of elements of $$L$$, of $$L$$ is a Lie subring but not necessarily an associative subring of this ring.

The $$(n+1)$$-Engel condition can be interpreted as saying that for any $$w \in L$$, $$(\operatorname{ad}(w))^{n+1} = 0$$. The results (1) and (2) can now be interpreted as results about sums of products in associative rings:


 * 1) $$\sum_{i=0}^n X^iYX^{n-i} = 0$$ where $$X = \operatorname{ad}(x), Y = \operatorname{ad}(y)$$.
 * 2) $$\sum_{j=0}^n (-1)^{j+1} \binom{n + 1}{j} X^jYX^{n-j} = 0 $$ where $$X = \operatorname{ad}(x), Y = \operatorname{ad}(y)$$.

Proof of (1)
Given: A $$(n+1)$$-Engel Lie ring $$L$$ with torsion-free threshold at least $$n$$. $$R$$ is the ring of endomorphisms of the additive group of $$L$$.

To prove': For all $$x,y \in L$$, $$\sum_{i=0}^n X^iYX^{n-i} = 0$$ where $$X = \operatorname{ad}(x), Y = \operatorname{ad}(y)$$.

Proof: