Bukhari

Hadith Commentary:
This refers to the period after the martyrdom of Abdullah ibn al-Zubayr (radi Allahu anhu), when the people had reached consensus upon the caliphate of Abd al-Malik ibn Marwan.

Hadith Commentary:
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In another narration, it is mentioned that after the martyrdom of Abdullah ibn Zubair radi Allahu anhu, when the people unanimously agreed upon the caliphate of Abdul Malik ibn Marwan, Abdullah ibn Umar radi Allahu anhu wrote him a letter, the contents of which were as follows:
“I acknowledge hearing and obedience to the servant of Allah, the Commander of the Faithful, Abdul Malik ibn Marwan, radi Allahu anhu, on the condition that his commands are in accordance with the Shari‘ah of Allah and the Sunnah of His Messenger sallallahu alayhi wa sallam.
I will adhere to them as much as I am able, and my sons also acknowledge this.”
(Sahih al-Bukhari, al-Ahkam, Hadith: 7205)

➋
Abdullah ibn Umar radi Allahu anhu would not pledge allegiance to anyone during times of turmoil; most likely for this reason, he did not pledge allegiance to either Ali radi Allahu anhu or Amir Muawiyah radi Allahu anhu.
When the people unanimously agreed upon Yazid ibn Muawiyah radi Allahu anhu, he pledged allegiance to him.
Similarly, when there was mutual discord between Abdullah ibn Zubair radi Allahu anhu and Marwan ibn Hakam, he remained aloof from both.
When, after the martyrdom of Abdullah ibn Zubair radi Allahu anhu, there was consensus on the caliphate of Abdul Malik ibn Marwan, he pledged allegiance to him.
In this, the condition is adherence to the Book and the Sunnah; for this reason, Imam Bukhari rahimahullah has mentioned this hadith here.
And Allah knows best.