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This does not work.

As of the November 10th, 2020 errata to the Sword Coast Adventurer's Guide, the green-flame blade cantrip (SCAG, p. 142; TCoE, p. 107) has a range of "Self (5-foot radius)".

However, the third benefit of the War Caster feat says (PHB, p. 170; emphasis mine):

When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.

Since green-flame blade has a range of self, it targets the caster, not "only that creature" which provoked the opportunity attack.

It should be noted that rules designer Jeremy Crawford has stated on Twitter than War Caster still works with booming blade, so it should still work here. I disagree that the rules clearly lend themselves to this interpretation, and think he should have made his rules clear enough to not need unofficial rulings on Twitter.

This does not work.

As of the November 10th, 2020 errata to the Sword Coast Adventurer's Guide, the green-flame blade cantrip (SCAG, p. 142; TCoE, p. 107) has a range of "Self (5-foot radius)".

However, the third benefit of the War Caster feat says (PHB, p. 170; emphasis mine):

When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.

Since green-flame blade has a range of self, it targets the caster, not "only that creature" which provoked the opportunity attack.

It should be noted that rules designer Jeremy Crawford has stated on Twitter that War Caster still works with green-flame blade. I disagree with this ruling, based on my explanation above, and think he should have made his rules clear enough to not need unofficial rulings on Twitter.

This does not work.

As of a November 10th, 2020 Errata to Sword Coast Adventurer's Guide, green flame blade has a range of self. War Caster says:

When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.

Since green flame blade has a range of self, it targets the caster, and not "only that creature" which provoked the opportunity attack.

It should be noted, rules designer Jeremy Crawford has stated on twitter than War Caster still works with booming blade, so should still work here. I disagree with his interpretation and think he should have made his rules clear enough to not need tweet rulings.

This does not work.

As of the November 10th, 2020 errata to the Sword Coast Adventurer's Guide, the green-flame blade cantrip (SCAG, p. 142; TCoE, p. 107) has a range of "Self (5-foot radius)".

However, the third benefit of the War Caster feat says (PHB, p. 170; emphasis mine):

When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.

Since green-flame blade has a range of self, it targets the caster, not "only that creature" which provoked the opportunity attack.

It should be noted that rules designer Jeremy Crawford has stated on Twitter than War Caster still works with booming blade, so it should still work here. I disagree that the rules clearly lend themselves to this interpretation, and think he should have made his rules clear enough to not need unofficial rulings on Twitter.

This does not work.

As of a November 10th, 2020 Errata to Sword Coast Adventurer's Guide, green flame blade has a range of self. War Caster says:

When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.

Since green flame blade has a range of self, it targets the caster, and not "only that creature" which provoked the opportunity attack.

This does not work.

As of a November 10th, 2020 Errata to Sword Coast Adventurer's Guide, green flame blade has a range of self. War Caster says:

When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.

Since green flame blade has a range of self, it targets the caster, and not "only that creature" which provoked the opportunity attack.

It should be noted, rules designer Jeremy Crawford has stated on twitter than War Caster still works with booming blade, so should still work here. I disagree with his interpretation and think he should have made his rules clear enough to not need tweet rulings.