(The hilarious mental image is Gaius having to deal with Fordola as a kid except then it’s not really hilarious because Fordola’d be a very angry, sad kid by the time she and her mother wound up under his aegis.) But that in itself led to another realization:

Gaius and Fordola use the same gunblade style. This could be passed off as standard training, except we see Regula van Hydrus use entirely different movesets in the Aetherochemical Research Facility, but uses the basics the same way, so this may be moot.
  • Fordola uses the same flame-patch AoE in the Stormblood 4.0 MSQ that Gaius uses in the Praetorium in 2.X MSQ.
  • Gaius uses the Terminus Est ability in both the Praetorium, and a much more powerful version in the 4.3 MSQ. Fordola uses this in the 4.0 MSQ vs. the WoL, and later the WoL and Lyse just prior to her capture.
  • Fordola does not hesitate to take the gunblade when offered it by Zenos, meaning she’s had training in its use.
  • Notable because: the Gunblade is a uniquely Garlean weapon (so we’ve seen so far, at least); Eorzeans under Garlean command tend to be given more brute-force magitech (Grynewalht, Rhitahtyn, NPCs in Garlean dungeons) or favour their own Eorzean weapons/magics. This reads as Fordola being given training by someone who no Garlean could really gainsay training her…
  • … and it fits Gaius’ pragmatism to train an angry Ala Mhigan kid whose father was his subordinate, and for whose death Gaius bears responsibility. Focus that anger towards training, towards the Ala Mhigans: outlet, and useful tool in one.
  • I’d have to look at the timeline to see if it actually works, though. Fordola’s 19 as of Stormblood, and seems to be 5-6 in the flashback; Gaius toppling Ala Mhigo was ~20 years before the current date.

To sum all of this up…
  • Fordola rem Lupis probably knows Gaius van Baelsar, aka The Black Wolf, fairly well.
  • Gaius may have felt somewhat responsible for Fordola when she was a kid, due to her father’s manner of death (essentially when he was a citizen of the Empire, protecting a future, productive citizen, and Gaius being a responsible Viceroy and canonically unbiased towards Eorzeans and very much a meritocrat)
  • He granted her (and her mother) citizenship and may have interceded to get Fordola trained on weapons she wouldn’t have been able to learn without him, if he didn’t do so himself.