Some people were confused about some of the jargon of the Elementals, so I typed up a little list (I'll add more by request):

First Variants (also known as "Terrans"): Earth Elementals. Powerful, bulky, dependable, and good at taking orders, Earth Elemental "Golems" are anything but subtle and incredibly blunt and straightforward. They worship the old Titans in Greek Mythology.

Second Variants (also known as "Flames"): Fire Elementals. Sachiko was one. Passionate, hot-tempered, reckless, and don't like being ordered around, especially by other Variants. Fire Elementals enjoy letting their emotions run wild, and have a particular conflict with Waters and Shadows (see below). They worship Hephaestus, the God of Fire in Greek Mythology.

Third Variants (also known as "Floods" or "Waters"): Water Elementals. In contrast to their Second Variant companions, Waters are cool, rational, collected beings. Their control over their judgments, which always overrule their passions, makes them completely opposite to their compulsive Second Variant counterparts. They worship Poseidon, the God of Water.

Fourth Variants (also known as "Gusts" or "Winds"): Wind Elementals. Aloof, without stress, these are the most laid-back of all the Variants. Without a care in the world, Fourth Variants are best given high-stress missions that no one else can take the pressure for. Like the Second Variants, they do little to control their emotions, not that they really need to. They worship Hermes, the God of Wind.

Fifth Variants (also known as "Shadows" or "Apparitions"): Shadow Elementals. Crono is the leader (the rank is "Elemental") of the Shadows. Shinji is an Acolyte (the equivelent of a Lieutenant). They are the eyes and ears of the Global Compact (the parallel world Crono and Sachiko come from). They are effectively the internal and external security branches of the Compact; think the FBI, CIA, NSA, and Dept. of Homeland Security (DHS) all rolled into one. Their strict control over their emotions is necessary for covert ops missions. They are forbidden to love or have powerful affection for anyone outside the Elemental Corps, lest they be killed or taken prisoner. They worship Hades, God of the Underworld.

Sixth Variants (Also known as "Temporals"): Space-time Elementals. None have been found by the Compact as of yet. However, the Shadow branch is making many recon missions to try and find these elusive Variants, for it is rumored that these Variants are incredibly powerful. It is not known what they worship, but it is suggested that they would worship Chronos (the God of time, no relation to Crono Blackwood).

As for the organizations that exist within the Genetic Lessons world...

The Global Compact:
Government: Democratic Republic
Population: App. 12,000,000,000 (12 billion)
Population growth: 3.8% every 10 years
Territory: Earth, as well as parts of the Moon and Mars
Capital: Athens
Most widely spoken language: Greek
Currency with highest value: Compact Dramal
Economic Model: Free-market Capitalism
Most widely practiced religion: Ancient Greek Mythology
Current Calendar Year: 2012 A.U. (After Unification)
Most highly respected occupation: Elemental Leader

This world exists as a parallel earth, a world based on the answer to the question "What if...". The tangent of this world to the Evangelion world goes back to the fourth century B.C., where Alexander the Great did not die as young as he did in the history texts. In fact, his empire continued to grow exponentially. By the time he finally did die many years later, he handed his empire over to many able-bodied lieutenants, who governed the Greek-Macedonian empire ably, and expanded it further. With the mental prowess of the Greek city-states under the command of the Greek-Macedonian empire, both military power and cultural envy led to a continual expansion and a constant upsurge in technology. Eventually, the entire world was in control of the Greek-Macedonian Empire, founding itself as the Global Compact. With a scientifically curious Athens-based culture instead of a scientifically static Rome-based Catholic Church, science and technology boomed exponentially.

The Global Compact is mostly self-governing; however, a six hundred man Assembly, whose members are chosen randomly among all citizens to serve a one-year term, oversees planetary crises and wars.

A few hundred years before the events that transpire in Genetic Lessons, a mutation is discovered in the population. The mutation itself is rare, with only a few dozen thousand popping up every year out of the nine billion Compact citizens. The mutation itself has five known Variants, each of which enables the host to unlock amazing powers to control nature (see Elementals above).

The Fundamentalist Imperium:

Government: Theocratic Monarchy
Population: App. 1,000,000,000 (One billion)
Population growth: -2.8% every 10 years (this figure is only accurate
in Fundamentalist-controlled areas; Americas and Australia not
discovered.)
Territory: Europe, Africa, Asia. Other continents unknown due to lack
of data.
Capital: Rome
Most widely spoken language: Latin (among aristocracy, clergy), Greek
(slaves, servants)
Currency with highest value: Gold (mostly barter among lower and
middle classes)
Economic model: Clergy-dominated Feudalism
Most widely practiced religion: Christianity
Current Calendar Year: 2296 A.D. (Anno Domini: The year of our Lord
Jesus Christ.)
Most highly respected occupation: Clergyman.

This is also a parallel earth, existing on the premise that Muhammad, the founder of Islam, simply never existed. Without the counterbalance that the rise of the Moslem Empire provided, the Roman-Christian Empire (brought together formally by Constantine) saw it's power and size grow by leaps and bounds as well. However, the Empire did not grow by combining the power of both culture and military: the growth of this empire was seeded exclusively in the latter. Pope-generals were a constant.

However, this expanding behemoth had one great advantage: a similar, yet different mutation, soon cropped up within the Fundamentalist Imperium, though much more prevalent among the Imperium's one billion people. The Imperium itself did not know what exactly had caused the change in many of their warriors, simply that their crudely-fashioned iron weapons and armor could be "blessed" by prayers, made by these genetically-enhanced holy warriors.

A Pope, centered in Rome at the Vatican, rules the Fundamentalist Imperium. The holy warriors of the Imperium are each specialized into various branches, such as Clerics that can cure maiming wounds, Paladins that fight with both blessed weaponry and destructive prayers, and the enormous Fanatics, born twice as large as any man and who know only how to kill heretics.