Increased Burning Damage

I tried to figure it out on the Wiki but iam still not sure and the Tooltip does not help either so i will ask here:

If i play an Elementist with Avatar of Fire with the Elemental Prolif Ascendency and i use a spell which has not a Burning Keyword, will Increased Burning Damage still work?

In my case i use a Lighning Spell + Call of the Brotherhood + Cold to Fire. So the Lightning Damage should be now all converted to fire and as Elementalist with the Buff i ignite most of the time with 100% Chance. If i now Add a Increased Burning Damage gem linked with the Lightning spell, will it increase the Burning damage the Spell causes with ignite?

Thank you a lot :)