Let $A$ be a cocommutative $K$-Hopf algebra, where $K$ is a field. Given Hopf subalgebras $X,Y$, one can define the commutator of $X,Y$ to be the subalgebra $[X,Y]$ of $A$ generated by the elements $$\{x,y\}=x_1y_1S(x_2)S(y_2),$$ where $x\in X$, $y\in Y$, $S$ is the antipode of $A$, and we are using Sweedler's notation. In the recent paper "A semi-abelian extension of a theorem by Takeuchi", it is showed that $[X,Y]$ is a Hopf subalgebra of $A$, normal if $X,Y$ are normal.
In the proof, to show that $[X,Y]$ is invariant under comultiplication $\Delta$, the following few lines of computation are presented:
$$\Delta(x_1y_1S(x_2)S(y_2))=x_1y_1S(x_4)S(y_4)\otimes x_2y_2S(x_3)S(y_3)\\ =x_1y_1S(x_2)S(y_2)\otimes x_3y_3S(x_4)S(y_4)\in [X,Y]\otimes [X,Y]. $$
I am not understanding this proof, and in particular its "shortness". I think I can prove the result, but I need at least one page of computation, based on the fact that as $A$ is cocommutative, we have the equality $(\Delta_{A\otimes_K A})\Delta=(\Delta\otimes\Delta)\Delta$. However, for sure there there is something clever that I am missing in the proof above, probably in the use of Sweedler's notation. Could you please give me some indication to understand why this proof work?