To help with the discussion later, you may want to add something about twists here, and something along the lines of 'it can sometimes be convenient for efficiency to do the computations of G2 in the twist Et, and so consider G2 to instead be a subgroup of Et.'
Comments from Chloe (Expert Reviewer)
To help with the discussion later, you may want to add something about twists here, and something along the lines of 'it can sometimes be convenient for efficiency to do the computations of G2 in the twist Et, and so consider G2 to instead be a subgroup of Et.'