Lie rings with a finite cyclic grading in which there are many commuting components

Let $L$ be a $(\mathbb Z/n\mathbb Z)$-graded Lie algebra (ring) with finite-dimensional (finite) zero-component of dimension $\dim L_0=r$ (of order $|L_0|=r$). If for some $m$, each grading component $L_k$ for $k\ne 0$ commutes with all but at most $m$ components, then $L$ has a soluble ideal of derived length bounded above in terms of $m$ and of codimension (index in the additive group) bounded above in terms of $n$ and $r$. If in addition $n$ is a prime, then $L$ has a nilpotent ideal of nilpotency class bounded above in terms of $m$ and of codimension (index in the additive group) bounded above in terms of $n$ and $r$. As an application, a corollary on metacyclic Frobenius groups of automorphisms is given.