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The item Helm of Brilliance includes a feature that reads (DMG, 173):

You can use an action and speak a command word to cause one weapon you are holding to burst into flames. [...] When you hit with an attack using the blazing weapon, the target takes an extra 1d6 fire damage.

The Great Weapon Fighting fighting style reads (PHB, 72):

When you roll a 1 or 2 on a damage die for an attack you make with a melee weapon that you are wielding with two hands, you can reroll the die and must use the new roll, even if the new roll is a 1 or a 2.

This question deals with which dice Great Weapon Fighting lets you reroll. According to the answer, it seems that the the feature was only intended to affect the weapon's damage dice, not extra dice from features like Sneak Attack or Divine Smite, or spells such as Hex. It seems that all of the weapon's damage dice, however, are affected, and so weapons such as a Frost Brand have their additional dice modified (1d6 cold damage in the case of Frost Brand).

In the case of the Helm of Brilliance's flame effect, can a 1 or 2 on the additional 1d6 fire damage be rerolled using Great Weapon Fighting?

On one hand, it seems like the extra damage from the Helm of Brilliance can't be rerolled, because it is clearly a spell-like effect rather than a natural part of the weapon. On the other hand, the effect clearly modifies the weapon rather than the attack, and so the additional damage could be considered a quality of the weapon.

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You can reroll the fire damage

Following the same logic as the question you linked you would be able to reroll the damage. The distinction is if the weapon is dealing additional damage or if you are adding damage to an attack.

My reading of the the Helm of Brilliance is that it turns the weapon into a blazing weapon. Thereby it is the weapon that deals additional damage, much like Frost Brand, though only while the effect is active.

I believe this effect falls into the same category as a spell like Shillelagh. These spells modify the base weapon damage rather than adding damage to your attacks like Hex. Others may disagree but I don't think there is any way that Great Weapon Master could not work with Shillelagh. Therefore it should work with the Helm of Brilliance.

As always, your DM will be the final arbiter but at my table I would allow it.

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