Let $f: H_1 \rightarrow H_2$ be a map of graded connected cocommutative Hopf algebras over a perfect field. Let $H \subset H_1$ be the Hopf algebra kernel of $f$, and let $I \subset H_1$ be the kernel of $f$, viewed as an algebra map. Let $\bar H$ be the positive dimensional part of $H$, and let $(\bar H) \subset H_1$ be the algebra ideal generated by $\bar H$. Clearly $(\bar H) \subseteq I$.

Questions: Does $(\bar H) = I$? It seems likely that this is a standard fact. If so, where is this in the literature?

(The hypotheses that the Hopf algebras are cocommutative, and having the field be perfect, just happen to hold in the situation I am considering, and perhaps are irrelevant.)