There is a fine line between arrogance and distance. In fact distance from people can be inherent to an arrogant behavior. But when distance is the result of a rational process instead of an ignorance-driven sense of self-importance, it can mean a healthy social and organizational behavior.
The main difference between arrogance and mere distance is the factor of deliberate attention seeking.
People sometimes take distance because they believe in the existence of an irreparable gap or difference with the others, and if taking distance would help prevent any future clashes, then distance means healthy, and should be respected. It is like telling the other person ‘I respect where you stand but I would rather not deal with it’.
Arrogance, on the other hand, is a deliberate attention seeking process to the self-perceived sense of superiority with a deliberate intention to ignore the other. It is like one would be doing everything to let people close and fascinated by them yet denying them the right to be close enough to them.
Although there is a debate on the fact whether distant leaders are as effective as close leaders, distance can still be a direction a leader would take when dubbed indispensable. Never the case for arrogance though, as it embeds a sense of disrespect to the other.
And in our relationships, it is important to identify what is what to avoid all misinterpretations and later frustrations.