DISCLAIMER: This guide is applicable to the current version of EBII (EB 2.0). It may not be applicable to future versions.
Authority traits
The main source of Authority in EBII will come from the 'factionleader' line of traits (Untested Leader, Established Leader, etc). The trait works by scripting various actions to give or take 'points' from the factionleader trait. At certain levels, the visible trait and effect will change, boosting or reducing your authority.
The + authority thresholds are as follows (Don't click the spoiler if you don't want to spoil it for yourself):
FactionLeader trait points can be increased if you:
-Have your faction heir win large battles *before* becoming faction leader. Authority and Factionleader points scale up to three large battles won. (The authority boost here is seperate from the factionleader trait, so it's a good way of getting extra authority.)
-Simply being a faction leader slowly boosts factionleader points slowly each turn.
-Win battles with an army led by your faction leader. The larger the battle, or the poorer your battle odds, the higher the boost. Even a minor skirmish or utter steamroll will give a small boost, however.
-Successfully completing a mission. (Think council missions from vanilla M2TW, or senate missions from vanilla Rome 1)
-Ransom captives taken in battle by an army led by your faction leader. Number of captives taken is irrelevant.
-Release captives taken in battle by an army lead by your faction leader. Number of captives taken is irrelevant.
-Destroy a faction. Does not need to be done by your faction leader, but has to be done by your faction's actions. (I think)
-Have a child. This is a one-time boost for your faction leader when he becomes a father.
-Have good traits (Sharp, Charismatic, Energetic, Optimistic). Charisma is the most important, intelligence in the middle; energy and temperament are least important.
-Have your faction leader close or inside your capital. This isn't very important and the boost is very small.
------------
FactionLeader trait points can go down as well, however. In addition, there is also a seperate line of traits that reduce authority permanently. These traits (Leadership Setback traits) cannot be countered.
The thresholds for Leadership Setback are as follows:
Leadership setback points are hard to get, though, so unless your faction leader screws things up badly (losing multiple battles) or has very bad personality traits (Dull/Uncharismatic/Langourous/Pessimistic) you don't have to worry about them.
You will lose factionleader trait points and gain leadership setback points if you:
-Have poor traits (Dull, Uncharismatic, Languorous, Pessimistic).
-Have little battle experience. Have your faction leader take part in at least two battles, otherwise this will pull down your authority quite quickly.
(The 'Blooded' trait occurs after first battle, so one more battle after getting the 'Blooded' trait.)
-Have an army led by your faction leader lose a battle. This one is a pretty significant hit.
-Fail a mission. (Think council missions from vanilla M2TW, or senate missions from vanilla Rome 1)
-Have your faction leader be away from the capital before amassing any factionleader points. This triggers when you have less than 3 factionleader points with a 10% chance.
Given that you get 1 factionleader point per turn, this can pretty much be ignored.
-Have your faction leader be away from the capital after amassing at least 3 factionleader points. This reduces factionleader points by 2 at an 8% chance per turn.
Given that you get 1 factionleader point per turn at a 100% chance, this can pretty much be ignored.
------------
The following ethnicities give +1 authority:
Carthage: Karthadastei
Makedonia: Antigonid
Epeiros: Aiakides
Seleukids: Seleukides
Ptolemaioi: Lagides
Hayastan: Yervanduni
Pontus: Pharnakides
Parthia: Arshaki/Arsaces (No bonus at first, gets +2 Authority as trait progresses. I'm guessing reforms tie into this.)
Parthia: Suren(No bonus at first, gets +1 Authority as trait progresses.)
Pritanoi: Pritanoi
Lusotannen: Lusotannen
Arevaci: Arevaci
Aruernoi: Aruernoi
Boioi: Boioi
Aedui: Aedui
Saka Rauka: Saka Rauka
Sauromatae: Saitae
Massylii: Massylian
Pergamon: Attalides
Kimmerios Bosporus: Leukonides
-----------------------------------------------------------------------------------------------------
Loyalty
This section covers the loyalty mechanics used by (almost) every faction. Certain factions have additional loyalty mechanics, but they are not covered here. Also note that each general starts with different initial loyalty stat, so some generals will be inherently more loyal than others.
To keep your generals loyal, boost your faction leader's command above 5 and influence above 6. Any generals with less than 5 command or 6 influence (if you meet the respective thresholds) will slowly gain the RespectFromCommand and RespectFromInfluence traits, each granting +1 loyalty.
In contrast, if your faction leader has less than 5 command or less than 6 influence, and any of your generals has >5 command or >6 influence, the general will get the AmbitionFromCommand and AmbitionFromInfluence traits, which reduce loyalty by 1 each; unless said general has 5 or more loyalty (before getting the Ambition traits).
------------
In addition, there are the court loyalty traits. Court loyalty applies to all factions except Rome, and to all generals apart from the faction leader and heir.
If your faction leader is in your capital or nearby (the file here states distance < 10, though I'm not sure what that means), and one of your generals is either in the capital or close by (distance < 10),
they have a 15% chance per turn to gain the CourtLoyalty trait, which boosts loyalty by 1.
However, if your faction leader is out of the capital or its surroundings (distance < 10) and a general is nearby or within the capital,
he has a 5% chance per turn of gaining the CourtIntrigue trait, which reduces loyalty by 1.
CourtIntrigue can be reduced by moving your faction leader back into the capital and having the general with CourtIntrigue stay there with him;
the trait will go away when CourtLoyalty fires.
CourtLoyalty does not go away if both your general and your faction leader travel away at the same time,
or if a general with CourtLoyalty travels away from the capital and your faction leader stays.
------------
Small factions get a loyalty boost, while large factions get a loyalty penalty. The larger the faction, the less loyal its generals.
------------
tl;dr Keep your FL's command above 5, influence above 6, and keep any generals apart from your faction heir away from the capital unless your faction leader is also there. Small factions are loyal, big factions disloyal.