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The elements shouldn't really reflect the eidolons that can be possible here, but they are showing some adjustment when it comes to element alignment. It's more about their Gestalt modes anyhow, since that's their real power.
Didn't Gilgamesh have a sword? I don't recall seeing the summon holding a weapon. It may be a complete redesign of an older summon like Remora for example. I'm hoping it isn't a new summon as I can't keep up with them all! (over the entire series I mean)
. . .Flying is simple. You just throw yourself at the ground and miss . . .