I'm pretty sure that Gaius Julius Caesar both command his army on foot and mounted. His horse were famous and brought courage to all who saw it. It was grey white but had some freak foots. In his early days Julius Caesar fought on foot but in the Gallic Campaign i'm sure he rode on his horse. He needed the overview but many times during the fights he may unmount himself to fight amongst the infantry in the battallions there was under pressure.
I'm also very sure that normal Roman Generals wasn't like Julius Caesar and under the hole battle sat on their horse to maintain overview. It's correct when the army march the general normally marchs with them on foot. But in battle it's on horse.
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