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

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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 Abelian ideal (?) of order p^k.

Related facts

Similar facts

Facts used

  1. Lower bound on order of maximal among abelian ideals in terms of order of nilpotent Lie ring
  2. 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.