The structure of connected (graded) Hopf algebras revisited

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