I believe I was able to compute the antipode of G with help from a very patient classmate using the Hopf Algebra diagram. But in Federico’s office hours he suggested that it would be easier to use Takeuchi’s formula. I attempted that and if I am using the formula correctly, he was right. I am not sure though if my notation is correct. If anyone has used Takeuchi’s formula and cares to compare notes please let me know.
Takeuchi’s Formula Vs. the Hopf Algebra diagram