How might emotional intelligence be manifest? I identify three things.

First: it is a question of value. This is when you start to place emotion higher in terms of what matters. Now feelings are not to be gotten over, ignored and put away in favour of the supposed right action. They don't get shunned in order to conform to the requirements of the world and its conventions. Such denial can bring about shame, being shamed by your feelings because they are deemed unacceptable the result of which is longer term damage to the psyche. Valuing emotions and appreciating their importance is the first step to avoiding the damage.

Second: to actually become more aware of feelings. This seems obvious because of course you know what you are feeling. But it is not necessarily true. Sometimes emotions are so repressed as to become distorted and appear back as something different entirely. Hatred of self can often project as hatred of another. So a more accurate awareness of feelings is required which depends on a degree of analysis and introspection. Names need to be put to emotions and a method of articulation found to express the inner dynamic.

Third: to integrate the first two, value and awareness, into a life. That is where the art of it comes in. It would involve being able to healthily process feelings in an appropriate way while still controlling actions. It could never do to always act on passion. The skill is to control without repressing, to be able to feel and act in a manner fitting to the moment.