Nasai

253. Commentary:

➊ The Sunnah method of ritual bath (ghusl) begins with ablution (wudu); therefore, after performing ghusl, there remains no need for a separate ablution, provided that during the ghusl, after performing ablution, he did not touch his front or back private parts with his hand. Otherwise, he will have to repeat the ablution.

➋ Similarly, if he did not perform the Sunnah method of ghusl, that is, he did not begin the ghusl with ablution, then even in this case, he will have to perform ablution after the ghusl.

430. Commentary: Since the prescribed (masnun) ritual bath (ghusl) begins with ablution (wudu), there is no need for ablution afterward, provided that during the bath one has not touched the private parts after performing ablution. If the ritual bath is not performed in the prescribed manner, that is, if it is done without ablution, then ablution will be required afterward. See also Hadith: 253 and its benefit.

1:
That is:
The ablution (wudu) performed during the ritual bath (ghusl) is sufficient.

Commentary:
(1)
The reason for this is that while performing the ritual bath (ghusl), they would first perform istinja (cleansing of the private parts) and then perform ablution (wudu). After that, they would not touch the concealed and specific body parts, and thus, they would perform the prayer with the same ablution used for the ritual bath.

(2)
Our esteemed researcher has declared the chain of transmission (isnad) of the aforementioned narration to be weak, whereas the issue mentioned in the narration itself is correct in essence. Most likely for this reason, other scholars have declared it to be hasan (good) and sahih (authentic). For details, see: (al-Mawsu‘ah al-Hadithiyyah, Musnad Imam Ahmad: 455, 454/40, Hadith: 24389)