Immediately a quote from the Brihadaranyaka Upanishad came to my mind. It's not "Know thyself" rather than "Love thyself" but I think it goes in a similar direction. In the very least, it may showcase why it's important to know and understand the self. It also underlines why seeking knowledge beyond the self may be deemed irrelevant.
Therefore, Yajnavalkya says to Maitreyi, “Nobody loves anything for its own sake.” All love is love of the Self, in the pure spiritual sense. Not this self or that self, myself or yourself, itself—this kind of self is not the point. It is the universal Self that is actually pulling you in some form, and you are not able to catch the point. There is an illusion that is presented to the sense organs, and under the impression—due to the delusion—you go to the object thinking that it is beautiful, that it is necessary, that it is meaningful. There is no meaning in anything in this world except the meaning of the Selfhood of that object. — Brihadaranyaka
In accordance to these teachings, knowing thyself will lead to knowing the other. In fact, I reckon one comes to a much deeper understanding of others if they have understood their own self first, rather than simply observing some second or third party.
That means, self-awareness must precede other-awareness. That's the correct/preferred/recommended sequence for awareness (self-awareness first, other-awareness second). However, the current understanding of consciousness (self & other awareness and more) is that other-awareness preceded self-awareness: humans and a few other animals that are/seem to be self-aware evolved very late in life's history given the evidence available to us. — TheMadFool
I do struggle to grasp the common understanding of self-awareness outside of a human context.
Wikipedia says: "While consciousness is being aware of one's environment and body and lifestyle, self-awareness is the recognition of that awareness."
But isn't being conscious in itself the awareness of consciousness? What I'm trying to say is that while I'm sure there is something like self-awareness, at the same time the idea we have of it seems rather artificial. Even the famous mirror test to me, seems to me more of a test of mental capacity, rather than self-awareness.
Consider then being conscious but not self-aware. You've just been told to "Know thyself" and so you go your philosophical quest to do just that. What is is that you do? Likely, you'll be exploring the concept of yourself and all the aspects of it. You'll do so by thinking, with your mind, constructing words in your head that seemingly fit your condition. By the power of the word and your imagination, you may even try to abstract yourself from your own mind and body, as if trying to take a third-person look on your first-person being. You come to one conclusion or the other, in terms of thinking got as far away from yourself as you could possibly get and then proclaim: "I know myself!"
Now my question is: Do you really become self-aware through that? Through putting abstract layer upon layer on your persona, ultimately deciding at some point "this view is an objective enough consideration of myself."? Then let us imagine a hypothetical person that knows no language, no words. If they are unable to articulate themselves as we do, can they never become self-aware?
Here's a theory of mine: While language is the greatest tool of mankind, it can sometimes become a curse. The use of words in everyday life has totally shifted the workings of our thoughts. Since language is inherited culturally, from a point of other-awareness so to say, it may represent a sort of barrier to truly and deeply understanding the self. I believe there's a more intrinsic, natural way of thinking and knowing. Thoughtless thinking, where no words are necessary. Why is it that those who practice meditation claim that to become self-aware, you must still the mind?
Similarly how you don't use words in your head to move your body, it seems obvious to me that they aren't a necessity for thought either. Rather, we may differentiate between two kinds of thoughts:
1. The ones I do not need to formulate. Those are the thoughts for the self. The thoughts the self knows.
2. The ones I do need to formulate. The self still knows these thoughts. However, these are meant to eventually be shared with others. Like how we discuss philosophy here, words for my thoughts are necessary so that I may share those contemplations. These formulated thoughts are for interaction. The issue here is: We learn this from such an early age and engage so much with others, that it becomes our normal mode of thinking.
So when the Self is engaged in thoughtless thinking, is that not the highest form of self-awareness? For when we think without words, we are most aware that the Self simply knows. The more I can rely on myself knowing that my Self knows, the more self-aware I become.
Also I believe that any living being is capable of this thoughtless thought. There are varying levels of complexity of course - but I think it goes hand in hand with consciousness itself. Thoughtless thought is what moves and maintains our body. It's what makes us dream at night. It's what lets an animal differentiate between predator and prey. And it's the silent voice that lets us know that our mind is chattering again when we are sat deep in meditation.
Certainly we have this ability from our birth. How can a baby learn anything if it's not capable of thinking without words? In that sense, I do believe that all life comes into being self-aware. Even the tiniest, seemingly dumbest organisms around us.
As you say, know thyself before you know the other. So perhaps it's just the other way around? Perhaps we're so awfully unaware of our self (by courtesy of blending words into our thought process), that we simply fail to understand the self-awareness of other beings?