Bukhari

Explanation:
The first handful (of water) is poured on the right side, the second handful on the left side, and the third handful on the middle of the head, as has been mentioned in chapter «من بدءبالحلاب اوالطيب». Imam Bukhari rahimahullah has indicated towards this very hadith here. And the relevance to the chapter heading is evident. According to some, the chapter heading is derived from «جمله ثم تاخذ بيدها على شقها الايمن», in which the pronoun refers back to the head; that is, then water is poured with the hand on the right side of the head, and with the other hand on the left side of the head. Kirmani said that the chapter heading is derived from this because the body, from the head to the feet, is included.

Hadith Commentary:
➊
Imam Bukhari rahimahullah, in order to establish his point, did not cite any marfu‘ (prophetic) hadith, but rather referenced the statement of Aisha radi Allahu anha.
According to Imam Bukhari rahimahullah, the statement of a Companion holds the status of a marfu‘ hadith.
This also indicates Imam Bukhari’s methodological position: that when a Companion says (كُنَّا نَفعَلُ) “We used to do,” it holds the rank of a marfu‘ hadith, whether or not the Companion attributes the action to the era of the Messenger of Allah sallallahu alayhi wa sallam.
Imam Hakim rahimahullah also holds this view, whereas other hadith scholars only grant the statement of a Companion the status of a marfu‘ hadith when it is attributed to the era of the Messenger of Allah sallallahu alayhi wa sallam.
(Fath al-Bari: 1/499)
➋
The purpose of this chapter heading is to show that, in ritual bath (ghusl), just as in ablution (wudu), the Sunnah method is to begin from the right side. However, an objection is raised that the chapter heading mentions the right side of the head, whereas the hadith cited mentions washing the right side of the body first.
In response, Allamah Kirmani rahimahullah writes that by “the right side of the body” is meant the right side of the body including the head; thus, the hadith aligns with the chapter heading, since the head is also part of the body.
(Sharh al-Kirmani: 2/140)
Hafiz Ibn Hajar rahimahullah has written that Imam Bukhari rahimahullah understood from the mention of washing the head three times in the hadith that the division is such that first the right side of the head is washed three times, then the left side three times, as has already been clarified in a previous hadith that the Prophet sallallahu alayhi wa sallam began washing from the right side of his head, then the left.
(Sahih al-Bukhari, al-Ghusl, Hadith: 258, and Fath al-Bari: 1/500)
However, in this hadith (no. 258), it is not definitively established whether what is meant is washing the head or applying fragrance.
If it is a case of washing the head, it is possible that he began from the right side of the head and simultaneously washed the right side of the body, then began from the left side of the head and included the left side of the body as well, but for the sake of brevity, the narrator did not mention the body and only narrated the words (فَبَدَأَ بِشِقِّ رَأْسِهِ الأَيْمَنِ) “He began with the right side of his head.”
➌
In this hadith, the right and left sides are mentioned in an absolute sense.
There is no restriction to the head, but Imam Bukhari rahimahullah, by restricting it to the head in the chapter heading, clarified that by “right side” and “left side” in the hadith, what is meant are the two sides of the head, because the method of washing the head is to take water in the palm and pour it on the right side of the head, and likewise wash the left side.
To wash the entire body, water is poured or both hands are used simultaneously.
Pouring water on the body one handful at a time is contrary to common practice.