With Rhys' permission, I thought that we should probably restrict the number of Githyanki and Githzerai secret societies to 9 because we don't want to get TOO large in this book.
Here's the ones that need detailed I can tell.
Duthka'gith: The Half-Dragons and Tiamat worshipers of the Gith. They're mostly Dragon Riders and have a lot of Prestige classes out of the Draconomicron. They're also something like societal outcasts from the Gith and I think it'd be cool if they've started seizing Githyanki Egg Farms to raise them into their faction.
Thus making them outlaws amongst Gith kind.
I imagine these Githyanki actually don't live on the Astral Plane but maintain their fortress on the Nine Hells.
The Kai'tarn'Gith: Roughly translated "The Warlords of the Gith." The Warlords are individual Githyanki Knights and Generals that have seized control of their own God Cities and Prime Material strongholds. The Kai'Tarn'Gith are all "Shoguns" and there's a curious honor amongst the various Gith groups. Specifically, its a conflict that isn't all out war.
Most of the Warlords are interacting as much through assassins, sabotage, skirmishes, subterfuge, bargaining, and diplomacy as they are with open warfare amongst each other. The ideal solution for most Githyanki Generals is they can force a rival into a subordinate position underneath them. I think we should detail two Githyanki Warlords as an example of their leadership.
I think the first one should be the one with the largest of the armies, the political mindset to unite the Githyanki, and also a willingness to ally with lesser races [In the interests of eventually conquering the universe). Sort of a Tokugawa Ieyasu typre figure. Very LE.
The next one, I think should be a die hard fanatic of the Githyanki with a huge army but whose mostly a danger because he's a fantic. He's all for attacking the Prime Material, burning it down, and starting to enslave other races as Gith dreamed. He's NE. He should also have (secret) ties to the Duthka'gith
The Silver Swords: This is a rough translation but these are a bunch of Githyanki Knights, some having switched over to LN or even LG of all things, that have devoted themselves to finding Gith to come back in their hour of greatest need. While some of them believe the legend literally, quite a few of the Knights believe that Gith is dead and needs to be brought back. While the tens of thousands of years since her death precludes a normal resurrection, they're confident that they can figure a way around it so long as they can find either a trace of her soul or her corpse.
The Silver Swords are starting to become Folk heroes of the Githyanki as they have no desire to involve themselves in the Civil War. Instead, they're mostly working to prevent their fellow Gith from suffering at the hands of traitors or protecting them from any soulless barbarians that would take advantage of the situation.
The Amakal'Kosh'Vlaakith: 'The Glorious Disciples of Vlaakith' are lead by mostly undead Githyanki mixed with Vlaakith's priests (their spells provided by Tiamat) that have been largely cast out of Githyanki society under the pretext of 'failing' to protect her. Most of the rank and file are Vlaakith's former secret police in the Ch'r'ai and her priests that believed her destined for Godhood.
The group maintains its fortress on the Negative Material Plane where it's all but unassailable and are mostly looking for where Vlaakith's soul was taken when her spirit receptacle was destroyed. They consider the majority of Githyanki to be traitors and are not above plots that cause large scale collateral damage along with using "barbarian" demons and devils to attack them.
It's good to have a group that's just unambiguously rotten to even the LE Githyanki.
Zerths: A Githzerai society that definitely need expanding.
Sha'Sal Khou: A mixed Githzerai and Githyanki secret society banned by both people and lead by the Githyanki Zetch'r'r that wants to reunite them into one race. The group claims responsibility for destroying the Lich Queen and quite a few believe them. I'm inclined to have Zetch'r'r killed and the group currently undergoing some serious problems at this time.
...Wait, there's a book now?
Just saying, my own ideas of the Githyanki vary considerably from yours. I'll continue posting my own content, so that others are free to agree or disagree with said ideas.
BoGr Guide to Missile Combat:
1) Equip a bow or crossbow.
2) Roll a natural 1 on d20.
3) ?????
4) Profit!