Izzy would have the subtype as long as you controlled it... from spellbook building on. If another Mage took control of Izzy somehow, then it would lose the subtype (and potentially gain a different one, depending on who the new controller is).
At no time does training ever cease to exists.
It's not a training issue, its an ability issue.
Step 4: Resolve Spell
At this point, the spell takes effect. The spell type (e.g.,
conjuration) and the text on the spell card determine the
effects of the spell. If you have cast an attack spell, you must
now resolve the attack.
RAW? I would say she absolutely does not gain additional trainings until she is resolved.
Her ability text is never RAW applied to her until after she is cast. It's not something like X = where the X is the actual casting cost and is explained later in the card. This is an ability, that technically should not exist until after she resolves. However, since she can be cast from lair / barracks we have to assume that this ability is special and works via the owner and additional schools, if applicable based on mage training, is always taken into account from an "owner" perspective.
Because, even though the control has changed, via reanimate, mind controller, or whatever, ownership does not.
All that could be cleared up if an X was in the subtype and then an ability text like, " If Izzy's owner is trained in.<insert school here>. X = <insert subtype here> " Owner is described in the codex and there is no issue since there is at least some precedent with the X = scenario.
It doesn't say that however and when I explained that to players, there is nothing in the rules to say she should work like that. There is no X in her subtype and there are no rules to support her being able to be cast from the lair / barracks aside from intent.
End of the day card abilities should trigger in the resolve spell set and not in the spellbook creation stage.
The simplest solution is to give her both subtypes and have her ability read "If the mage is not trained in war she loses the soldier subtype" and so on.
Either way,
These are not the droids we're looking for; Move along.
#endRant