The ninja master who trained Snake Eyes and Storm Shadow is the Hard Master.
The Hard Master is the head of the Arashikage ninja clan and a central figure in the origin of both characters. In the Marvel Comics continuity, he is Storm Shadow’s uncle and trains him before taking Snake Eyes in as a student. Snake Eyes eventually becomes the Hard Master’s favored pupil and is chosen as his successor, intensifying the rivalry between the two students.
The Hard Master’s death becomes one of G.I. Joe’s defining mysteries. An assassin hired by Cobra kills him with an arrow intended for Snake Eyes, while evidence falsely points to Storm Shadow. Later stories reveal that Zartan was responsible. The Soft Master and Blind Master are related Arashikage figures, but neither is the master specifically identified as training both Snake Eyes and Storm Shadow.