I'm looking for a finite-dimensional Hopf algebra (over any field) that is unimodular, has unimodular dual, but is not involutory. Is there such a thing?


Here's what I know:


* By Radford's formula, the antipode $S$ must have order 4.
* Suzuki [has constructed](https://www.jstor.org/stable/43685968) unimodular Hopf algebras over any field that are not involutory (and where $S^2$ is not even inner). My calculations so far suggest that these are not counimodular, but I still need to double-check.
* It is well-known that semisimplicity implies unimodularity. Therefore finding a cosemisimple unimodular Hopf algebra that is not involutory would be even stronger (or dually, a semisimple counimodular one).
* A [result](https://arxiv.org/abs/q-alg/9712033) of Etingof and Gelaki is that "semisimple cosemisimple" implies involutory, so this combination would be too strong.



Hopf algebra that is unimodular and counimodular but not involutory
It turns out that an explicit example of such a Hopf algebra already appears at the end of Radford's seminal paper [The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite](https://www.jstor.org/stable/2373888) as Example 2.


The example is a bit too complicated to reproduce here, so let me just note that it is 8-dimensional, can be defined and has the desired properties over any field, and happens to be isomorphic to its opposite.


