I want to prove that the cocommutative finite dimensional Hopf algebras over an algebraically closed field of characteristic zero are group algebras (for some finite group) and that the commutative f.d. Hopf algebras (over an algebraically closed field of characteristic zero) are dual to group Hopf algebras (for some finite group). More precisely, I want to prove the following proposition:

Let $k$ an algebraically closed field of characteristic zero and $Η$ a finite dimensional $k$-Hopf algebra. Then, we have the following Hopf algebra isomorphisms:

- If $Η$ is commutative, then: $Η \cong (kG)^{*}$ (for some finite group $G$).
- If $Η$ is cocommutative, then: $Η \cong kG$ (for some finite group $G$).

I understand that the above result can be extracted as a consequence of the the Cartier-Konstant-Milnor-Moore classification theorem, according to which, a cocommutative $k$-Hopf algebra is isomorphic to a smash product between the universal enveloping algebra of the Lie algebra of the primitives of $H$ and the group Hopf algebra of the grouplikes: $H\cong U(P(H))\sharp kG(H)$. ($k$ is of course considered alg. closed with zero characterictic).

However, I want to provide an independent proof. After working a bit on it, I have devised a proof (which I am posting below as an answer) which however makes use of a later result, the so-called Larson-Radford theorem. Could there be some different approach (involving or not the Larson-Radford theorem)?