Shaykh Al-Albaani, in one of his books "The Prophet's Prayer" says that: “He, sallallaahu 'alayhi wa sallam, used to send prayers on himself in the first Tashahhud as well as the other. He also established it for his Ummah, ordering them to send prayers on him after sending peace on him. Which means we have to say "atahiyatu lilah...and Allahumma sali ‘ala Muhammad..." in each Tashahhud! Please, I’d like to know more about it, because not a lot of people know, and I think it is very important praying according to the Sunnah of Prophet Muhammad (SAS)

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Some scholars hold the view that it is needed to recite the whole Tashahhud including the prayers on the Prophet after the 2nd Rak'ah of the Thuhr, ‘Asr, Maghrib and ‘Isha prayers for the general evidence stated in the Prophetic Hadeeth. Other scholars believe that one should recite the Tashahhud only, i.e. without saying the prayers on the Prophet after the second Rak'ah as mentioned above.

Imam Ahmad  and Ibn Khuzaymah reported from Ibn Mas'ood that he described the Prophet's prayer saying: “When the Prophet completes the first two Rak'ahs, he pronounces the Tashahhud and then resumes standing. And when he is in the last Rak'ah, he says the whole Tashahhud including the prayers on the Prophet and supplicates with what he likes, and then pronounces the Salaam.' Ibn al-Qayyim said: “There is no authentic narration that proves that the Prophet said the prayers on himself and his family in the first Tashahhud, nor sought refuge with Allah from the torment of the grave, the Hellfire, and from the ordeal of the this life, death and anti-Christ. Then, if some scholars believe that it is desirable to follow the first stated opinion (reciting the whole Tashahhud after the 2nd Rak'ah) that is, in fact, a deducted understanding which, according to the sound opinion, is fixed only after the final Tashahhud. Thus, we say it is better to avoid such difference of opinion although it is permitted, since both opinions are within the Ijitihad which is acceptable and both are according to the Sunnah. So, Muslims have to respect each other's opinion.” [End of quote]

For more benefit, please refer to Fataawa 355897 and 364543.

Allah knows best.