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We construct the universal enveloping preassociative and postassociativealgebra for a pre-Lie and a postLie algebra respectively. We show that thepairs $(\mathrm{preLie},\mathrm{preAs})$ and$(\mathrm{postLie},\mathrm{postAs})$ are Poincare-Birkhoff-Witt-pairs, for thefirst one it's a reproof of the result of V. Dotsenko and P. Tamaroff.

Poincare-Birkhoff-Witt theorem for pre-Lie and postLie algebras