I just had a peek at the chapter on Self-enquiry (practice). It says we should stay in the I-awareness. Only if unable to do so, or taken out of it, we can use techniques such as question "who the I" is who is associating with a person.
Since at that moment we are lost as the personality, a mental questioning of this personality seems okay. Otherwise, if we can immediately come back to awareness, that is the best.
Constantly remaining in or reverting to awareness will reduce vasana's (mental tendencies), in the end the Self will destroy what is left.