Ibn Khuzaymah

Sayyiduna Abu Hurairah (may Allah be pleased with him) narrates that when the Messenger of Allah (peace and blessings be upon him) would stand for prayer, he would say « اللهُ أَكْبَرُ » while standing, then when he would bow (ruku‘), he would say « اللهُ أَكْبَرُ », then when he would straighten his back after ruku‘, he would say « سَمِعَ اللهُ لِمَنْ حَمِدَهُ » “Whoever praises and glorifies Allah, Allah has heard his voice.” And then while standing he would say « رَبَّنَا وَلَكَ الحَمْدُ », then when he would go down for prostration (sajdah), he would say « اللهُ أَكْبَرُ », then when he would raise his head from prostration, he would say « اللهُ أَكْبَرُ », then when he would perform the (second) prostration, he would say « اللهُ أَكْبَرُ ». Then when he would raise his head, he would say « اللهُ أَكْبَرُ ». Then the Prophet (peace and blessings be upon him) would continue in this manner throughout the entire prayer until he completed it. And when he would rise after the tashahhud in two rak‘ahs, he would say « اللهُ أَكْبَرُ ». Then Sayyiduna Abu Hurairah (may Allah be pleased with him) said: I am the one who performs prayer most similar to the prayer of the Messenger of Allah (peace and blessings be upon him).

The Prophet ﷺ did not say "Allahu Akbar" every time he stood up, but sometimes he would say it. When he ﷺ raised his head from bowing (ruku'), he would say "Allahu Akbar," but except for raising his head from bowing, he ﷺ would say "Allahu Akbar" every time he stood up.