Possible minor spoilers ahead
The first war academy was founded in 3125 AU by King Elijah I of Gwynland, exactly a decade after the end of the Burning Revolt.
His goal was to train a large standing army, in case of an eventual retaliation by the Solis Empire. A retaliation that fortunately never came, because it would undoubtedly succeed. The war academy did manage to raise a professional army, but the Crown was unable to maintain it for long.
Discredited by the nobility, King Elijah changed his goal from a large standing army to a smaller elite force made up of the sons and daughters of the largest houses, which proved to be quite successful.
Today, there’s a war academy in each of the ten duchies —each managed by its Duke/Duchess with some minor support from the Crown—, one in the Heartlands —managed by the Queen/King consort— and one in the Royal Duchy of Heresburg, managed by the Royal Heir.
There are Seven Elemental Branches of Magic taught in the war academies: pyromancy, hydromancy, aeromancy, geomancy, luminomancy, umbramancy, and fulminomancy.
Pyromancy is the branch focused on creating, controlling and shooting fire. As with Fulminomancy, its spells are mostly destruction oriented while lacking the versatility of other branches. Also similar to Fulminomancy, they are mostly used as artillery. Pyromancy’s opposite is hydromancy.
Hydromancy is the branch focused on the manipulation of water and sometimes ice. In combat, it is the most situational of branches as one can’t create water and unlike air, earth or stone, there may be times where there isn’t any around to use. What makes hydromancers invaluable in an army is their ability to heal wounds. Hydromancy’s opposite is pyromancy.
Aeromancy focuses on the manipulation of wind. In combat, they can lower the air resistance around themselves, or do the opposite to their enemies, create deadly blades of wind or even fly for short periods. It’s opposite is geomancy.
Geomancy focuses on controlling earth and stones. They can sense the subtlest of earth pulses to act as scouts alongside aeromancers, or sap enemy fortifications. They’re most commonly used as vanguards, shielding the rest of the troops from the enemy’s artillery. Its opposite is aeromancy.
Luminomancy is the branch focused on creating and manipulating light. They’re able to create focused beams of light which, although lacking in power against similar attacks from pyro or fulminomancers, make up for it by easier and less costly spells. Through their mending spells, they’re able to provide first-aid similar to hydromancer’s healing which, again, lack in effectivity but are easier to use. Illusions are also possible for more skilled luminomancers. It’s opposite is umbramancy.
Umbramancy focuses on creating, materializing and manipulating shadows. They’re the most versatile of branches, comparable to hydromancers with the advantage of being capable of operating almost anywhere. It’s opposite is luminomancy.
Fulminomancy focuses on creating lightning, the hardest element to control. Due to that, it is the one with the least amount of spells among all elemental branches of magic. It's also impossible to hold a fulminomancy spell for more than a few seconds. It is the only elemental branch of magic without an opposite branch, with the consensus being that it comes from the heavens and nothing can oppose something divine.
Branches of magic not taught in the academies:
Necromancy is a branch of magic focused on controlling the dead. When casting spells, necromancers emit dark mists, which leads to them being —erroneously— grouped up with umbramancers. Regardless of nationality or social class, the view of necromancy as evil is one of the few things people tend to agree on. As such, despite its possible applications, especially in warfare, the practice of necromancy is a crime punishable by death, and it isn't taught in the academies for obvious reasons.
Divination is a branch of magic focused on attempting to see the future. Such as necromancy and umbramancy, divination is mistakenly grouped with luminomancy due to them emitting bright lights when casting it. By definition, it is an imperfect practice due to the future not being set in stone, and at any time being capable of going in an incalculable number of paths. Divination is different from other branches in the sense that it’s impossible to use it to directly harm people — other than the one casting it. Often times diviners go mad when attempting to see either too far or the infinite different futures possible. It isn’t taught in the academies due to it being too vague and unreliable.