Rashi citing Targum Yonasan: It is in order to stoke coals of fire, like we find "va'Yishlach ha'Keruv Es Yado

Pesachim 119a: The Kesiv refers to Hashem's hand. It is under their wings in order to receive Teshuvah of penitents out of 'sight' of Midas ha'Din (which would protest).

Radak: The Kesiv teaches that its hand is like a human hand, as if it wrote v'Yadav Yedei Adam. This is like "Einayich Yonim" (Shir ha'Shirim 1:15) - they are like doves' eyes.

Rashi: In the four directions of the world, for they have faces and wings in every direction ¹ .

Malbim: Just like the wings used to fly are fixed on all four sides, by every face, to be able to fly in any direction, so the hands are fixed on all four sides.

Malbim (according to the metaphor): On every side that they raise wings to rise above the Ma'arachah, whether for Chesed, Din or mercy, their actions are with choice and desire. They are not forced via the Ma'arachah, rather, according to Hashem's desire.

Malbim according to the Rambam: The two arms were for all four faces of each Chayah, just like each has only two legs. From where the arms join to the body and below, they are one body; only above the arms they separate to four faces.

Rashi citing Targum Yonasan: They are the same for all four of them.

Malbim: Their wings are with their faces; there are wings for every face. We need u'Fneihem