# Milnor-Moore and the characteristic zero homology of H-spaces

In their work on Hopf-algebras:
https://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf>
on the last page p.263, they say that the Hurewicz map $$\lambda: \pi_*(G)\otimes K \rightarrow H_*(G,K)$$ is morphism of Lie-algebras for fields $$K$$ with characteristic $$0$$, which is fine. But they claim that the image of this map is excatly $$P(H_*(G,K))$$ the primitive elements of the Lie-algebra $$H_*(G,K)$$. For the proof he refers to Cartan and Serre without giving an reference. Does anyone know if the proof is written down anywhere?
• Such an element must come from $H_*(S,K)$, but all elements of $H_*(S,K)$ are primitive. Jun 11, 2019 at 17:59
This is immediate from Proposition 9.2.4 in the book $$$$More concise algebraic topology'' by Kate Ponto and myself. No originality is claimed.