Flameblade Angel — ruling 1

If a source deals damage to you and/or one or more permanents you control at the same time, Flameblade Angel’s last ability will trigger that many times. For example, if a creature with trample deals damage to a blocking creature you control and you, Flameblade Angel’s ability will trigger two times.

Flameblade Angel — ruling 2

Note that the last ability is optional. Say two players each control a Flameblade Angel, and a source controlled by the first player deals damage to the second player or a permanent they control. The resulting triggered ability may deal damage to the first player, causing the ability of that player’s Flameblade Angel to trigger. That ability may deal damage to the second player, and so on. This cycle will repeat until the game ends or one player declines to use the ability, putting an end to the carnage.

Flameblade Angel : règles, fonctionnement et interactions (MTG)

Flameblade Angel is one of those cards players keep asking about. Here is, in plain words, what its official text says, the rulings published by Wizards of the Coast, and the combos players actually run.

Official card text

Flying Whenever a source an opponent controls deals damage to you or a permanent you control, you may have this creature deal 1 damage to that source's controller.

How it works : Flameblade Angel

Original English text — Wizards of the Coast / Scryfall

#1April 8, 2016· Wizards of the Coast
If a source deals damage to you and/or one or more permanents you control at the same time, Flameblade Angel’s last ability will trigger that many times. For example, if a creature with trample deals damage to a blocking creature you control and you, Flameblade Angel’s ability will trigger two times.
No other card is named in this ruling.
#2April 8, 2016· Wizards of the Coast
Note that the last ability is optional. Say two players each control a Flameblade Angel, and a source controlled by the first player deals damage to the second player or a permanent they control. The resulting triggered ability may deal damage to the first player, causing the ability of that player’s Flameblade Angel to trigger. That ability may deal damage to the second player, and so on. This cycle will repeat until the game ends or one player declines to use the ability, putting an end to the carnage.
No other card is named in this ruling.