CFI A&P
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I have to disagree. I don't see anywhere in the regulations stating that the training HAS to be done under part 135/121. I've read this section many times and don't believe it says anything of the sorts. Also, the FAA Office of Chief Council issued issued a letter dated July 2016 about this very thing. It clearly states that a properly qualified and typed part 91 PIC may do ALL the training required by 61.559(b) as well as the required endorsements and signing of 8710. The same PIC is authorized to do all of this even without having any sort of CFI.
https://www.faa.gov/about/office_or.../2016/Axton - (2016) Legal Interpretation.pdf
So unless there is something I'm missing, its not sketchy at all. It is perfectly legal to do so. I'm not going to argue whether it is prudent or not, but it is definitely legal.
That link leads to a 404 not found page.
At the moment I am unable to post the references, many of them refer to definitions. However in a nutshell: ATPs may instruct while in engaged in air transportation service [61.167(a)(2)] (Part 91 is not defined as an air transportation service). SIC training does not have to be under 121/135 but if the training is being conducted under 61/91, then the person providing the training must be an instructor. This is how The staff at a 142 sim center may not be CFIs but they are permitted to instruct based on their ATP and the authority of the training center.
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