When Stone of Jordan is worn, all static elemental damage bonuses (displayed in the details window) should be the same. Spectral Blade - Flame Blades increases fire damage bonus, it seems to be the static fire damage bonus, the value in the details window is increased, Chantodo’s Resolve changes its element, but all other elemental damage bonuses stay unaffected. The same effect can be observed with Black Hole - Absolute Zero. On the other hand, another skill that also gives a temporary bonus to one element - Wave of Force - Arcane Attunement, behaves consistently - isn’t shown in the details window, doesn’t work with SoJ, etc.
If it’s not a static elemental damage bonus, then Flame Blades shouldn’t affect other skills and items depending on static elemental bonuses, like Chantodo’s Resolve. If it’s a static elemental damage bonus, then Flame Blades should work correctly with Stone of Jordan and increase all elemental damage bonuses.
On the video the character starts attacking with Flame Blades while wearing Stone of Jordan. Only fire damage bonus increases. Note that the video also showcases another, unrelated issue with Enchantress’ Amplification skill, described in another bug report.
Video: SoJ + Flame Blades - YouTube