To write down some thoughts about Ixi aka Travier
Instead of a ghost father shenanigan, he will suffer from a split personality with Ixi being the first and major character while Socraes Travier being the enforced one, reforged by the Emperor.
He will suffer from time to time regarding flashbacks or such until he faces off against the agents of chaos and the already hinted warp rift encounter.
What exactly happened is unknown so we got much to cover in offscreen tales like stories, etc.
During this encounter, Ixi shows up again, battling a mental fight against Travier or having a trip with him, showing the truth of their shared past (the events around his discovery, etc.)
In doing so, Travier ceased to exist and Ixi is back again.
Those brothers close to him, have a relation with Travier, not with Ixi. This is crucial for certain relations like Gwal and Socraes.
Difference between him and Lorgar - two ideologies
While Lorgar became the chosen prophet of the pantheon due to his belief, Ixi joins their cause for a slightly different reason: restoring the natural order.
The universe was born from chaos, thus establishing a natural order, a balance between the mortal realm and the warp, both being the mirror of each other. Both relying on each other. Mortal life is a reflection of their warp presence/ soul and visa ve.
There were others in the past, mighty species and empires which tried to deny the warp. The Old Ones, the Necrontyr, the Eldar - they all failed. Their realms crumbled while chaos persisted. Thus, Ixi came to the conclusion, the Imperium of Man will fall in time if it does not accept chaos.
So all he's trying to do (in his own twisted mind) is to save mankind by fusing it with chaos/ convincing it to accept the warp and its entities. Here we got a nice opposite to Andezo. One tries to save by embracing and one tries to destroy by annihilate. The one convincing the other becomes his greatest enemy.
This ties to his attempts to improve his sons, etc. Even the Aztec culture, with slight adjustments, suits this cause, as well.
Just some notes I wanted to save before they get lost on discord.