Existence of abelian ideals of small prime power order in nilpotent Lie ring

Statement
Suppose $$p$$ is a prime number and $$G$$ is a nilpotent Lie ring of order $$p^n$$. Then, if $$k$$ is a nonnegative integer such that $$n \ge 1 + k(k-1)/2$$ (i.e., $$n > k(k-1)/2$$), $$G$$ has an fact about::abelian ideal of order $$p^k$$.

Similar facts

 * Existence of abelian normal subgroups of small prime power order (see more facts related to that)

Facts used

 * 1) uses::Lower bound on order of maximal among abelian ideals in terms of order of nilpotent Lie ring
 * 2) uses::Finite nilpotent Lie ring implies every ideal contains ideals of all orders dividing its order

Proof
Outline: We use Fact (1) to show that there is an abelian ideal of order at least $$p^k$$, and then use Fact (2) to find within that an abelian ideal of order exactly $$p^k$$.