Upvote:2
This site Documents Required for Domestic Travel in Air India says about ID documents:
Below are the documents required for domestic travel in Air India
An ID card (Passport, Voter Photo Identity card issued by Election Commission of India, Aadhaar or m-Aadhaar issued by Unique Identification Authority of India (UIDAI), PAN Card issued by Income Tax Department, Driving Licence issued by RTO, Service Photo Identity Card issued by Central / State Government, Public Sector Undertakings, local bodies or Public Limited Companies, Student Photo Identity Cards issued by Government Institutions/ Government recognized Educational Institutions, Nationalised Bank Passbook with attested Photograph, Pension card / Pension documents having photograph of the passenger, Disability Photo ID Card / handicapped medical certificate issued by the respective State / UT Governments / Administrations).
(The original source of this information at Air India was found by @NicolasFormichella)
So if your ID is photo ID and your private school is recognised by the Government as such, you should be OK. They also accept certain bank photo ID.
But the best thing is to contact the airline itself.