NOTE: This is my own personal theory that I am concluding based off of experience and my knowledge of the socionics system. Additionally, I am not claiming that I am the only one to notice or discover this, and I am sure that theories like this exist under different names or under differing contexts. These are simply my own observations not to be taken as fact, and I am well aware that I will be prone to deep criticism and disagreement.
I believe that within every single quadra, there is a black sheep that by experience, has struggled to relate to the general quadra descriptions. Those types are SEI of Alpha, IEI of Beta, ILI of Gamma, and LSE of Delta. I can justify each of them with the supposed quadra descriptions and what beliefs are contrary in comparison.
Sites I will be using to compare:
Wikisocion's Type Descriptions (Gulenko and Stratiyevskaya)
The Alpha quadra as a whole seems to be focused very much on idea synthesis, intellectualism, and gradual, systematic change that is universally logical and appealing (hence why the alpha quadra is often compared to the intellectual Enlightenment of the 17th to 18th centuries). ILE and LII are the general intellectual arbiters of their ideas, with ILE as the one to put out their own innovations while the LII helps form it into a logical, universal cohesive. The ESE has its own role within this as a 'room brightener' to the heat created by the ideas. After all, cold logic and concepts are unappealing without some means to bring emotional warmth and comfort to the table. The SEI may find themselves confused in all of this, as they merely have 1D Ne and are more keen to focusing on creating physical atmospheres of glee and bliss with their 4D Si without over-complicating matters with logical cohesions or overvaluation of what is true and pure. The over-analytical nature of the alpha NTs may be unappealing for them as an observer. This is not to go against the SEI at all, and they still hold this sort of childlike innocence most of the time by mere valuation of Ne, and they are generally very open-minded, optimistic people, hence why they are in the alpha quadra.
The Beta quadra seems to describe eponymous gatherings as loud and rambunctious gatherings of comradery, especially as a means to plan for something over the top and tremendous in nature. The EIE acts as the emotional inspiration and passionate 'cult leader' who is able to lull and empathize with people to create a powerful force. The SLE and LSI are the sword and shield respectively of any kind of movement, with the SLE being able to hit the hardest and not be hesitant to do so, and the LSI defending movements with rigor and idealism. The IEI sort of misses out on all of this, especially with their 1D Se and 4D Ni. They tend to overthink and avoid being impulsive in these contexts. They are likely the wallflowers in these rambunctious gatherings, and may not appeal to extreme or otherwise barbaric tactics that types like SLE and EIE may consider. However, they're still beta for the reasons of desiring emotional, passionate mediums of expression, romanticization of ideas that may be considered avant-garde or even "edgy", and their general interest in sociopolitical issues & where society is heading in the long term.
The Gamma quadra is focused very much on in-and-out groups, competition, efficiency, and social-climbing. After all, this is the quadra most associated with 'capitalism', both the good and bad of it. LIE is the stereotypical entrepreneurial type who generally has the best ideas in promoting efficiency in a descriptive manner. The SEE is the go-getter type, nicknamed "The Politician" by many, and is able to appeal to their audience and form emotional bonds over mutual goals. The ESI acts as a 'support' most of the time, and as an emotional check and balance to the LIE and SEE to make sure certain things are not being overdone. That leaves the ILI, who generally does not relate to this venture-capitalist mission, and would rather spend time in contemplation, writing down their thoughts on these societal subjects, perhaps even in criticism. Generally, they do not seem to be moral arbiters either (like the ESI or SEE), as they have been known to promote ideas such as egoism. However, their rugged individualism, their selectivity with friends, as well as having a 'matter of fact' way of perceiving reality does keep them in touch with Gamma enough by a thread.
The Delta quadra generally prefers peaceful, minimalist settings where they can work on their own projects, and develop relations with people in a harmonic, mature manner. The EII is the prime example of this, possessing an inner seniority but also this gentle, almost childlike grace in their interactions. The IEE is generally carefree with a powerful inner child, letting them tinker with ideas in their minds that would have been lost in age. The SLI contributes to this too, and while a little more gruff than the delta NFs, still desiring an atmosphere of pleasure and realism. The delta environment has been likened to a forest or an isolated setting with close friends. However, the LSE, with its 1D Fi, may find this unappealing. It's not that they are incapable of being good people, or dislike forming intimate relations with others, but rather their 4D Te may be focused on more entrepreneurial matters, and their decision-making may conflict with the delta NF's moralism and humanism. The possibility of forming a house in the woods in comparison to a spiffy office space is a baffling idea to consider for an LSE. However, the LSE still possesses delta qualities such as the instinct against virulent or poetic passion, the dislike of external interruption/distraction, and the ability to self-improve as an autonomous entity.