PranAm, Amrut..
However I am very sure that this is commented by Adi Shankar only. I am not seeing anything wrong in his commentary. So what's the question of doubting its authenticity related to him?
Jeeva is said as part through subtle body of mind and intellect. Check, The verse 5.8 is mentioning subtle body of Jiva, which is mentioned in the verse as Mind and Intellect.
In the next verse, to show its sharp subtleness it is imagined as part of hair. However the verse at end says that this jiva itself becomes Infinite.
So I think this wouldn't be the reason to dispute over his authorship.
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