We show that for a given nilpotent Lie algebra $\mathfrak{ g}$ with $Z(\mathfrak{ g})\subseteq [\mathfrak{ g},\mathfrak{ g}]$ all commutative post-Lie algebra structures, or CPA-structures, on $\mathfrak{ g}$ are complete. This means that all left and all right multiplication operators in the algebra are nilpotent. Then we study CPA-structures on free-nilpotent Lie algebras $F_{ g,c}$ and discover a strong relationship to solving systems of linear equations of type $[x,u]+[y,v]=0$ for generator pairs $x,y\in F_{ g,c}$. We use results of Remeslennikov and St\"ohr concerning these equations to prove that, for certain $g$ and $c$, the free-nilpotent Lie algebra $F_{ g,c}$ has only central CPA-structures.
Categories: math.RA