By default Adepts do the same thing that Magicians do in order to initiate. No matter the Awakened individual, it requires an Extended Arcana+Intuition test with a Threshold of the Grade of Initiation you are trying to reach, with an interval of 1 month. Presumably, this test represents the person studying the depths of Magic and how it relates to them. This relates to all forms of Awakened. Yes, Adepts are less likely to use Arcana for other things, but the basics all relate back the same way.
Adept Ways, Traditions, Mentors, and even Ordeals to a small extent just shape what the individual might do to perform this Magic/self exploration.
I do want to add in that Ordeals don't (usually) replace the requisite Arcana test to complete the initiation, so not all Ordeals actually change what you might do to complete your initiation, they would just add more on top of it.