In "M. Kaneko and D. Zagier, A generalized Jacobi theta function and quasimodular forms, Prog. Math. 129, 165-172 (1995)" there is a proposition stating essentially that $E_2$, $E_4$ and $E_6$ are algebraically independent.
Unfortunately there is no proof there. I know how to show the algebraic independence of $E_4$ and $E_6$.
How does the proof for $E_2$ work?