The Clifford action is not faithful on "S": Cl"n"C can be identified with End("S") ⊕ End("S"′), where "u" acts with the opposite sign on "S"′. More precisely, the two representations are related by the parity involution "α" of Cl"n"C (also known as the principal automorphism), which is the identity on the even subalgebra, and minus the identity on the odd part of Cl"n"C. In other words, there is a linear isomorphism from "S" to "S"′, which identifies the action of "A" in Cl"n"C on "S" with the action of "α"("A") on "S"′.