So this dvijātmaja, son of a brāhmaṇa. He's not addressed as a brāhmaṇa because his actions are different. Unless one acts as a brāhmaṇa, he's not accepted as a brāhmaṇa. This is the Vedic system. Lakṣaṇam.
- yasya yal lakṣaṇaṁ proktaṁ
- puṁso varṇābhivyañjakam
- yad anyatrāpi dṛśyeta
- tat tenaiva vinirdiśet
- (SB 7.11.35)
Lakṣaṇam. Brāhmaṇa's lakṣaṇam, kṣatriya's lakṣaṇam. Lakṣaṇam means symptoms. One must act as a brāhmaṇa. So he is accepted as brahmātmaja. Of course, everyone can claim that his father is like that. But that does not mean he has acquired the father's qualifications. That is not meaning. And onward the son may claim that "Such and such big man is my father." Of course, from social point of view, we sometimes give respect, as it will be seen. Aśvatthāmā, because he happened to be a great personality, Droṇācārya, so he was excused from capital punishment. Otherwise, Arjuna decided to kill him. But because he was a very great man's son, guru-putra, considering that "If he dies, then guru's wife will be very much unhappy," Draupadī advised... She was unhappy on account of death of her sons. So they considered that Aśvatthāmā's mother, she's innocent, but on account of the son's death, she would be unhappy. So considering all these points he was excused from the capital punishment.