The aayah contains ismullaahi al-aa’tham (Allaah’s Most Greatest Name) affirmed in His Saying:
[ الحيُّ الَقيُّوم ]
Allaah, there is no god worthy of worship except He, the one who possesses the perfect living, and Who is established on His Own, Self-subsisting, and by Whom all things subsist.
These two Names were mentioned in three places in the Qur’aan, in the two soorahs known as az-zahraawayn  (AlBaqarah and Aal-’Imran), and in Soorat Taha.
The people of ‘ilm (ahlul-‘ilm) said: The Most Great Name of Allaah is only in these two Names since they comprise the Most Beautiful Names in their entirety, and hence the attribute of Perfection is implied in the Name [ الحيُّ ]and the attribute of ihsaan (beauty) is entailed in the Name: [ الَقيُّوم]
 Allaah is the one who sustains, protects, prepares, and runs the affairs of all things as He wills in accordance with His Knowledge, Wisdom, and Justice.
 Zahraawayn (dual form of zahra): The bright ones. The Prophet (صَلَّى اللّٰهُ عَلَيْهِ وَسَلَّم) said,
“Recite the Qur’aan, for on the Day of Resurrection it will come as an intercessor for those who recite it. Recite the two bright ones, Al-Baqarah and Soorat Aal-’Imraan, for on the Day of Resurrection they will come as two clouds or two shades, or two flocks of birds in ranks, pleading for those who recite them. Recite Soorat Al-Baqarah, for to take recourse to it is a blessing and to give it up is a cause of grief, and the magicians cannot confront it.”
Reported in Saheeh Muslim (English Translation), vol. 2, no. 1757.
 In Al-Baqarah (2:255), in Aal-’Imraan (3:2), and in Taha (20:111).
Abu Umaamah (رضي االله عنه) reported that the Prophet (صَلَّى اللّٰهُ عَلَيْهِ وَسَلَّم) said,
“Allaah’s Most Great Name is in three soorahs in the Qur’aan, in Soorat Al-Baqarah and in Aal-’Imraan and in Soorat Taha.”
Al-Qaasim bin ‘Abdur-Rahmaan, one of the narrators of this hadeeth, searched the Qur’aan and found
Aaytul-Kursi in Soorat Al-Baqarah (2:255):
[ االلهُ لا إله إلاَّ ُ هوَ الحَيُّ القَيُّوم ]
And in Soorat Aal-’Imraan (3:2):
[ اللَّهُ لَا إِلَٰهَ إِلَّا هُوَ الْحَيُّ الْقَيُّومُ ]
And in Soorat Taha (20:111)
[ وَعَنَتِ الْوُجُوهُ لِلْحَيِّ الْقَيُّومِ ]
Reported by al-Haakim, Ibn Maajah, at-Tabarani, at-Tahaawi, and others. Shaykh al-Albaani authenticated its isnaad (As-Silsilah As-Saheeha), vol. 2, no. 746.
Source : From the eBook: Tafseer Ayatul-Kursi – By Shaykh Muhammad bin Saalih Al-’Uthaymeen. With Related Points of Benefit From the Shaykh’s Commentary on Imaam ibn Taymeeyah’s Sharhul ’Aqeedatil-Waasitiyyah, Prepared by Dr. Saleh As-Saleh (rahimahullah)
Read the below excellent article:
Al Hayy, Al-Qayyoum: Two of Allaah’s names which comprise the perfection of Attributes and Actions – Shaykh Uthaymeen | Dr. Saleh as-Saleh