On [this](https://en.wikipedia.org/wiki/Shuffle_algebra) wikipedia page is stated that over the rational numbers, the shuffle algebra (over a set $X$) is isomorphic to the polynomial algebra in the Lyndon words (on $X$). I was wondering if you can show me a proof of this or give me a reference. Ideally I am looking for an explicit isomorphism between these two algebras.



The shuffle algebra over the rationals is isomorphic to the polynomial algebra in the Lyndon words
This is proven in Theorem 6.3.4 of [Hopf Algebras in Combinatorics](https://arxiv.org/abs/1409.8356) by Grinberg and Reiner.


They call it "Radford’s theorem on the shuffle algebra" citing Theorem 3.1.1(e) of [A natural ring basis for the shuffle algebra and an application to group schemes](https://doi.org/10.1016/0021-8693(79)90171-6) by Radford where you can find another proof.


