Working over an arbitrary base scheme S, we define the canonical quadratic pair on the Clifford algebra associated to an Azumaya algebra with quadratic pair. Given an Azumaya algebra \(\cal{A}\) with quadratic pair, i.e., with an orthogonal involution and a semi-trace, its associated Clifford algebra’s canonical involution is only orthogonal in certain cases, namely when \(\text{deg}(\cal{A})\) is divisible by 8 or when both 2 = 0 over S and \(\text{deg}(\cal{A})\) is divisible by 4. When \(\text{deg}(\cal{A})\geq 8\) , our definition of the canonical quadratic pair on the Clifford algebra is extended from previous work of Dolphin and Quéguiner-Mathieu, who worked over fields of characteristic 2. When \(\text{deg}(\cal{A})= 4\) , we show that no canonical quadratic pair exists.
