2-Engel Lie ring implies third member of lower central series is in 3-torsion

Statement

Suppose $L$ is a Lie ring that is a 2-Engel Lie ring, i.e., $[x, [x, y]] = 0$ for all $x, y \in L$ . Note that, by equivalence of definitions of 2-Engel Lie ring, this is equivalent to saying that $[x, [y, z]] = [y, [z, x]] = [z, [x, y]]$ for all $x, y, z \in L$ .

Then, the third member of the lower central series of $L$ , i.e., $[L, [L, L]]$ , is in the 3-torsion of $L$ . In other words, $3 [L, [L, L]] = 0$ .

Given: A Lie ring $L$ such that $[x, [y, z]] = [y, [z, x]] = [z, [x, y]]$ for all $x, y, z \in L$ .

To prove: $3 [x, [y, z]] = 0$ for all $x, y, z \in L$ .

Proof: The proof follows by plugging the given data into Jacobi's identity, which states that:

$[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0$ for all $x, y, z \in L$