Lie ring of given right-normed Engel-type signature

Definition
Suppose $$a_1,a_2,\dots,a_m$$ are positive integers. A Lie ring $$L$$ has right-normed Engel signature $$(a_1,a_2,\dots,a_m)$$ if it satisfies the following:

$$[x_1,x_1,\dots,x_1,x_2,x_2,\dots,x_2,\dots,x_m,x_m,\dots,x_m,x_{m+1}] = 0 \ \forall \ x_1,x_2,\dots,x_m,x_{m+1} \in L$$

where the bracket is right-normed, and each $$x_i$$ occurs $$a_i$$ times.

In shorthand, we call such a Lie ring a $$(a_1,a_2,\dots,a_m)$$-Engel-type Lie ring.

When $$m = 1$$, then having the right-normed Engel signature $$(a_1)$$ is equivalent to being an $$a_1$$-Engel Lie ring.