According to the original meaning of self-awareness over 99.9 % of humanity qualify for this trait because people are aware of their existence. So, this is not enough to be considered special. However, self-awareness has a new meaning along with a new claim that it distinguishes leaders from others, according to Daniel Goleman. 
Self awareness starts with the idea of recognizing and regulating emotions. But, how frequently have we seen leader spewing angry words (or tweets) only to later regret their words? They express emotions powerfully, but have little ability to regulate them. Frequently they fail to understanding the impact of their own internal motivation on themselves.
How important is control over your own emotions? If you feel vexed due to a person who chatters constantly with minimal output, you may decide to be self-aware and not speak your mind because they might be hurt by your critical words. Even children eventually learn to mask irritation. So, it would seem this skill not hard to learn. However, we see some leaders confronting others in public with an openly emotional barrage of hostility. They have not figured out how to conceal feelings when faced by an annoying threat to their status.
Leaders may recognize that the polite route is not the best way to get others moving. An irate tone is a good motivator, so people use it too much. But this has a hidden cost for those in positions of power. Those under them become disengaged, demotivated, and very skilled at avoiding confrontation. After the initial rush to respond to the leader’s demand, they tend to find a corner to hide in and get good at looking busy.
The delicate balance between expressing feelings that are “negative” so people know they have fallen short of expectations, and saying words that are harmful may be impossible to achieve. A person may learn to regulate the expression of emotions, but cannot regulate how sensitive or oblivious people are to what is said. People may take offense to a statement that is not even noticed by others. However, sensitive people who easily take offense to criticism can also use this to manipulate other people.
Concealing emotions comes at the cost of consuming energy. This all leads to the conclusion that people can be too self-aware, paying constant attention to how others respond and burning out through attempts to regulate how they appear. In reality a low level of self-awareness is often evident in leaders.  The cost of self-awareness may be one of the reasons why those desiring to be leaders have decided to toss this quality aside.
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