The structure of connected (graded) Hopf algebras revisited

Let $H$ be a connected graded Hopf algebra over a field of characteristic zero and $K$ an arbitrary graded Hopf subalgebra of $H$. We show that there is a family of homogeneous elements of $H$ and a total order on the index set that satisfy several desirable conditions, which reveal some interesting connections between $H$ and $K$. As one of its consequences, we see that $H$ is a graded iterated Hopf Ore extension of $K$ of derivation type provided that $H$ is of finite Gelfand-Kirillov dimension. The main tool of this work is Lyndon words, along the idea developed by Lu, Shen and the second-named author in [24].