Tirmidhi

English Commentary:
1:
The Ka'bah consists of four corners (arkān).
The first corner possesses two virtues:
First, that the Black Stone (Hajar al-Aswad) is installed in it; second, that it is established upon the foundations of Ibrahim (qawāʿid Ibrāhīm). The second corner possesses only the second virtue, i.e., it too is established upon the foundations of Ibrahim. The Shami (Levantine) and Iraqi corners possess neither of these two virtues;
these two are not established upon the foundations of Ibrahim. Therefore, the first is to be kissed (taqbeel), and the second is only to be touched (istilām), and for the remaining two, neither kissing nor touching is prescribed.
This is the opinion of the majority (jumhūr).
However, some people consider kissing the Yemeni corner (Rukn Yamani) to be recommended (mustahabb),
which is not prohibited.

2:
Imam Shafi'i rahimahullah has answered the statement of Mu'awiyah radi Allahu anhu by saying that not touching (istilām) the Shami and Iraqi corners does not mean abandoning them; the pilgrim is circumambulating (performing tawaf) around them, he is not abandoning them.
Rather, in both action and omission, this is following the Sunnah.
If not touching these two corners is considered abandoning them, then not touching the spaces between the corners would also be considered abandoning them, whereas no one holds this view.

Hadith Commentary: Benefits and Issues: The Black Stone (Hajar al-Aswad) and the Yemeni Corner (Rukn Yamani) are referred to together (taghleeban) as the "two Yemeni corners." Since both of these are upon the Abrahamic foundations, they are to be touched (istilam). The Black Stone possesses a double virtue; therefore, it is not only touched by hand but, by unanimous agreement, it is Sunnah to kiss it. Although some of the noble Companions (radi Allahu anhum ajma'in) used to perform istilam of all four corners, none of the Imams accepted this practice.

615 Lexical Explanation:
«يستلم» means to touch with the hand. This is Sunnah in every circumambulation (tawaf).
«غير الركنين اليمانيين» is an abbreviation of the "ya" after the letter "noon," and sometimes it is also recited with tashdeed (emphasis). It is attributed to the direction of Yemen. Since their orientation is towards Yemen, they are called "Rukn Yamani." And by «ركن البيت» is meant one of its sides. By these two corners is meant the corner with the Black Stone (Hajar al-Aswad) and the southwestern corner of the Ka'bah. The reason for touching (istilam) these two corners is that they are established upon the foundations laid by Ibrahim alayhis salam. As for «ركنين شاميين», that is, the corners on the side of the land of Sham (Greater Syria), they do not have this status.