We study post-Lie algebra structures on (g, n) for nilpotent Lie algebras. First we show that if g is nilpotent such that H 0 (g, n) = 0, then also n must be nilpotent, of bounded class. For post-Lie algebra structures x · y on pairs of 2-step nilpotent Lie algebras (g, n) we give necessary and sufficient conditions such that x • y = 1 2 (x · y + y · x) defines a CPA-structure on g, or on n. As a corollary we obtain that every LR-structure on a Heisenberg Lie algebra of dimension n ≥ 5 is complete. Finally we classify all post-Lie algebra structures on (g, n) for g ∼ = n ∼ = n 3 , where n 3 is the 3-dimensional Heisenberg Lie algebra.

We give an explicit description of commutative post-Lie algebra structures on some classes of nilpotent Lie algebras. For non-metabelian filiform nilpotent Lie algebras and Lie algebras of strictly upper-triangular matrices we show that all CPA-structures are associative and induce an associated Poisson-admissible algebra.

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