Is a 13-year-old who killed a child while in a state of extreme anger held accountable?

If you killed a child while you were not an adult, then there is no retaliation (qisas) upon you, because the pen is lifted from the boy (meaning he is not held accountable), and it is considered an accidental killing for which blood money (diyah) is due from your 'aqilah (male relatives on your father's side). However, if you were an adult at the time of the killing, then the rulings for an intentional killer apply to you, including retaliation (qisas) or blood money (diyah) and expiation (kaffarah), and you must repent to Allah.