Let $H$ be a Hopf algebra and $K$ a Hopf subalgebra of $H$. If $H$ is finite-dimensional, then by the Nichols-Zoeller Theorem $H$ is free as a left (and right) module over $K$.

Moreover, the same conclusion holds when $H$ is pointed, by the main theorem of [D. Radford: Pointed Hopf algebras are free over Hopf subalgebras, J. Algebra 45, 266-273 (1977)], available here

http://www.sciencedirect.com/science/article/pii/002186937790326X

In both the previous cases, can one always find a basis $\mathcal{B}$ of $H$ over $K$ with $1\in \mathcal{B}$?