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SAE International, as a standards organization, maintains several alloy numbering systems, one of which, for steel grades, is the SAE steel grades system.
In the 1930s and 1940s the American Iron and Steel Institute (AISI) and SAE were both involved in efforts to standardize such a numbering system for steels. These efforts were similar and overlapped significantly. For several decades the systems were united into a joint system designated the AISI/SAE steel grades. In 1995 the AISI turned over future maintenance of the system to SAE because the AISI never wrote any of the specifications.
Today steel quotes and certifications commonly make reference to both SAE and AISI, not always with precise differentiation. For example, in the alloy/grade field, a cert might say "4140", "AISI 4140", or "SAE 4140", and in most light-industrial applications any of the above is accepted as adequate, and considered equivalent, for the job at hand, as long as the specific specification called out by the designer (for example, "4140 bar per ASTM-A108" or "4140 bar per AMS 6349") is certified to on the certificate. The alloy number is simply a general classifier, whereas it is the specification itself that narrows down the steel to a very specific standard.