Shankara's Gita Bhasya says that the Self transmigrates. Is this true?

Question

Advaitins mostly believe that the Self doesn't transmigrate and that it's the subtle body (known as jiva/jivatma) that actually does. But then in Shankara's Gita bhasya it is mentioned that the self changes bodies like we change garments. Below is a translation of Bhagavad Gita 2.22 (the original text can be seen here):

22. Just as a man casts off worn-out clothes and puts on others which are new, so the embodied (Self) casts off worn-out bodies and enters others which are new.

Here is Adi Shankara's commentary (translated by Alladi Mahadeva Shastri):

Just as, in this world, a man casts off the clothes that have been worn-out and puts on others which are new, in the same manner, like the man (of the world), the embodied Self abandons old bodies, and, without undergoing any change, enters others which are new.

Here's a diagram of the Self dwelling in the innermost region. Is this what Shankara meant in his verse, that this in-dwelling Self travels along with the five sheaths? Or did he mean something else?

Answer 1

Shankaracharya uses Sanskrit word "dehi" in his commentary which is translated as "embodied self" (not self or Atman) and it indicates Jiva. You can read bhashya in Sanskrit from Gitasupersite of IITK.

Sanskrit Commentary By Sri Shankaracharya:

।।2.22।। वासांसि वस्त्राणि जीर्णानि दुर्बलतां गतानि यथा लोके विहाय परित्यज्य नवानि अभिनवानि गृह्णाति उपादत्ते नरः पुरुषः अपराणि अन्यानि तथा तद्वदेव शरीराणि विहाय जीर्णानि अन्यानि संयाति संगच्छति नवानि देही आत्मा पुरुषवत् अविक्रिय एवेत्यर्थः।। कस्मात् अविक्रिय एवेति आह

Translation of Sanskrit commentary by Swami Gambirananda.

2.22 Yatha, as in the world; vihaya, after rejecting jirnani, wornout; vasamsi, clothes; narah, a man grhnati, takes up; aparani, other; navani, new ones; tatha, likewise, in that very manner; vihaya, after rejecting; jirnani, wornout; sarirani, bodies; dehi, the embodied one, the Self which is surely unchanging like the man (in the example); samyati, unites with; anyani, other; navani, new ones. This is meaning.