A second attribute of emotional intelligence is self-management. Lack of self management can result in blowing up at people, shouting, or committing sexual harassment. Another aspect of self-management is to motivate yourself, especially when it comes to doing something you really don't want to do. It is not easy to confront difficult people, or to say things that must be said to co-workers, but a good self manager buckles down and does unpleasant but necessary tasks. Emotional intelligence also involves awareness of others, and the ability to discern the likely effect your words or conduct will have on someone else.
Social skills are an important part of emotional intelligence.
One of the most important aspects of social skills, I think, is the ability to influence others in such a way as to preserve the relationship when the influencing is done. We often underestimate the importance of emotional intelligence in the success of lawyers, judges, and, yes, professors. We also assume, I think, that emotional intelligence is an innate characteristic that cannot be taught - at least after early childhood. But I believe emotional intelligence can be taught in the same way other job skills can be taught. Some individuals will be more capable learners in these areas than others, but all can learn.