By dissecting the self, I very much mean analyzing institutions, really the environment in general, and how that environment creates the individuals that we are as it interacts with our genetic structures. What we are looking at there are structures of identity and behavior.
The self is also analyzed in the phenomenological sense, in the entity that we postulated at the center of recurring thoughts, at the center of our head, as that which streams the flow of perceptions. I follow people like Thomas Metzinger (The Ego Tunnel) and Bruce Hood (The Self Illusion) among many other cognitive scientists and philosophers (etc.) who focus on the idea that the concept of self that we naively postulated in the past does not match with our present best understanding. Some of what we engulfed in the sense of self was posited there because of transparent structures of brain to mind. We could say something like the little voice that runs through your head, your immediate experiencing of world and body and thoughts, encouraged us to postulate an entity, the self, that is no longer holding up under the closer inspection of neuroscience. Before the arising of cognitive science, broadly speaking, there were thoughtful criticizers from philosophy, psychology, and Buddhism.
There is often wrangling around the term illusion, which should often just be ignored in cases where conversations break down into such. There are a few good things to keep in mind. When people say the self does not exist or that it is an illusion, they often are arguing against its conceptualization as humans have created it from various experiences. There is usually reference to some continuous, seamless, unified and unitary entity that we postulate as who we are. As we dig deeper into the brain we are finding something that is different than many of those things. It makes some sense to me that we would call our original postulations illusions, as they were arrived at out of naive reflections on our experiences. There are other senses of self, of what is usually conceptualized as what it means to be a self, that are not illusions. We are brains and bodies, and if by self we just mean that generally persistent entity encompassing such, including much of the persistent brain/mind structure, then that meaning of self is not an illusion.
One of the main themes around this subject is that our mental thoughts were not structured to be faithful analyzers of what those mental thoughts are, and thus eventually what human beings postulated as the concept of self naturally consumes misleading information. On a similar structure, we wan look at the phenomenology of making a choice, of having contemplations and desires running through our head as we actively select among various options, without having any access to the working of the brain and much of the other constraining and thought-organizing factors. That structure all means that we experience an unconstrained, unlimited control of our choices from the first person perspective. I think it is one of the reasons why compatibilism will always be empty. It makes far more sense that our concept of free will is wrapped up in this constant experience of choice making, and that this experience has a very misleading or even self-unanalyzable phenomenological aspect. As to the general problem, R. Scott Bakker at the Three Pound Brain gives good arguments to these themes of misleading phenomenology, and he even goes further to claim that such a gap is unbridgeable. We are doomed to bad theories in some ways. I am not that pessimistic. I think we will eventually get a grasp on consciousness and these other mental phenomena.
I advocate, or playfully enjoy, cleaning up language and concepts. There are always equally adequate ways of getting across the same concept, and making those changes allows us to control connotation as we try to get to what is important. So, with that, I am happy to discard the self and free will to the dust bin. What's important is trying to figure out what kind of creatures that we are, what our experiences consist in, and in what kind of selves and society we wish to create.