Ahh, I hate just jumping into discussions like this, but I wanted to give kudos for someone finally recognizing that, considering the information we've been given in the story, Elysium simply is not Sophitia. From what I understood of the story, Elysium is the spirit of Soul Calibur, who took on the visual appearance of Sophitia to better manipulate our hero, Patroklos. His idealization of his mother, seeing her as a wise and benevolent figure, along with his obsession to be reunited with his family and right the wrongs that have been made against them, made him completely willing to bend over backwards to achieve whatever she asked of him. His loyalty to his family and his own self-delusions prevented him from seeing past Elysium's familiar face for the majority of the game - something which he wasn't able to truly get past until the very last moments of the story, even after his transformation into Alpha Patroklos. There's a reason Elysium was the final boss of the game - she had only a single purpose that needed fulfilled, and she would stop at nothing to achieve it, no matter what the cost. Lives were of no consequence to her, so long as Soul Edge was destroyed.
As Pat put it, "Our mother would never pit her children against one another."
... anyway. Back to Pyrrha now. :3