I don't know of any other elements, but I just want to point out that it's hard to really explicitly describe "elements of $\text{Gal}~\bar{\mathbb{Q}}/\mathbb{Q}$"; see e.g. [this answer][1] for a much clearer explanation than I could provide.


  [1]: https://mathoverflow.net/questions/2791/understanding-gal-bar-q-q/2812#2812