The ruling on greeting the one who is praying, and the Sunnah regarding how to respond to it during prayer.

The Shaykh : For example, regarding greeting the one who is praying, those who hold that this act is disliked have no explicit evidence from the Sunnah, let alone from the Book. Rather, they argue for it based on understanding and opinion, and all of that is subject to error. Meanwhile, there are authentic, established hadiths from the Prophet (peace and blessings be upon him) that he was greeted while he was praying on more than one occasion, and that he responded to the greeting offered to him without objecting to it even once. For example, in Sahih Muslim, there is the hadith of Jabir ibn 'Abd Allah al-Ansari (may Allah be pleased with him) that the Prophet (peace and blessings be upon him) sent him on an errand. When Jabir returned, he found the Prophet (peace be upon him) praying, so he offered him the greeting, and the Prophet (peace be upon him) responded with a gesture of his head.
There is another hadith like this one, containing another benefit that is not objectionable for you to hear. It is the hadith of 'Abd Allah ibn Mas'ud (may Allah be pleased with him), who was among the Companions who migrated to Abyssinia and was absent from the Prophet (peace and blessings be upon him) for a period of time. When he returned, the first time he met the Prophet (peace and blessings be upon him) was while he was praying, so he offered him the greeting. The Prophet (peace be upon him) did not respond with words, but rather, again, with a gesture of his head. Ibn Mas'ud said: "I was seized by what was near and what was far." Meaning, Ibn Mas'ud began to think deeply, considering things from every angle. He wondered why the Messenger of Allah (peace and blessings be upon him) did not respond to his greeting, given that his prior experience with the Prophet (peace and blessings be upon him) before traveling to Abyssinia was that the Prophet (peace and blessings be upon him) used to respond to greetings during prayer with words. In the early days of Islam, if someone greeted the one who was praying, he would respond with words while praying. This was a permissible matter. And so, Ibn Mas'ud left the Prophet (peace and blessings be upon him) for Abyssinia. Therefore, something troubled him in his heart when the Messenger (peace be upon him) did not greet him with the verbal greeting he was accustomed to receiving from him. He began to think that perhaps he had made a mistake with the Messenger (peace be upon him) and he had become angry with him. When the Prophet (peace and blessings be upon him) finished his prayer, and without doubt he knew, either through his wisdom (peace be upon him) or through the revelation of his Lord, what was troubling Ibn Mas'ud's heart, he said to him, answering what was troubling his heart: . So, if a man greets the one who is praying and he responds to the greeting, then... this was abrogated. It was permissible before, but then it was abrogated. At that point, Ibn Mas'ud's heart was at peace, and he knew that there was nothing to be concerned about, but rather it was the Shari'ah of Allah (Mighty and Majestic) descending as the wisdom of Allah (Mighty and Majestic) requires. In this hadith, as well, is the confirmation by the Messenger (peace be upon him) of offering the greeting to him while he was praying, along with the abrogation of the one who is praying responding to the greeting of the one who greeted him, which had previously been prescribed.
Similarly, for example, it is reported in Sunan Abu Dawud from the hadith of 'Abd Allah ibn 'Umar that the Prophet (peace and blessings be upon him) visited the Ansar in Quba at their mosque. They began to come to him in succession, and they would find him in prayer. They would offer him the greeting, and he would respond to their greeting with a gesture of his hand. One of the narrators of this hadith described this gesture, saying: "He would place the palm of his hand toward the ground and the back of his hand toward the sky (peace and blessings be upon him) while he was praying." So, the one entering the mosque would say to him: "Peace be upon you, O Messenger of Allah." And the Prophet (peace and blessings be upon him) would do nothing but do this: "Peace be upon you," raising his hand like this, with the palm of the hand toward the ground and the back of the hand toward the sky.
So, we take from these hadiths—and, praise be to Allah, they are all authentic—two matters. The first: the permissibility of offering the greeting to the one who is praying. The second: the obligation of responding to the greeting with a gesture of the hand or the head. And this—namely, the variation by the Messenger (peace be upon him) in the gesture, using either the hand or the head—contains wisdom. That is because gesturing with the head is, without doubt, more appropriate, as it is a movement, whereas raising the hand like this... when the one who is praying hears the greeting, a person feels that he is substituting the gesture of the head for the gesture of the hand, and he is content with that. This is because prayer is built upon tranquility, calmness, and stillness. Therefore, it is not permissible to fidget in prayer or to perform movements in it that Allah (Mighty and Majestic) has not prescribed through the tongue of His Prophet (peace and blessings be upon him).
It was the custom of some of the Companions in the early days of Islam that when they exited the prayer with the taslim, they would wave their hands like this, saying "Peace be upon you and the mercy of Allah, peace be upon you and the mercy of Allah." So, he (peace and blessings be upon him) said to them: . The horses, the sun, meaning... the two horses playing with their tails. He is likening these hands when they are raised to the tail of a horse when it raises it while... This means he disapproved of this action, and he confirmed that by saying: . Therefore, the principle is that the one who is praying does not perform a movement unless there is a need for it or it is prescribed, as we are currently discussing. So, if someone greets you and there are no other people praying around you, for example, then merely doing this makes it understood that you have finished the matter. No, you are among a group where your response to the greeting by gesturing with the head is not visible. In that case, you raise your hand, and the raising should also be to the extent that achieves the purpose of the raising, which is to make it understood by the one who greeted you that you have received his greeting and that you have responded to it.